Properties of Functions

Even and Odd Functions

A function is even if its output is unchanged under negation of the input, for every

x

in its domain:

Even Function Test

f(-x) = f(x) \quad \text{for all } x \in \text{Dom}(f)

Learn more about this formula: Even Function Test →

Browse all functions formulas →

The graph is symmetric about the

y

-axis — the left half mirrors the right.

The function

f(x) = x^2

is even:

f(-x) = (-x)^2 = x^2 = f(x)

. So are

f(x) = |x|

,

f(x) = x^4

, and

f(x) = \cos(x)

.

A function is odd if its output is negated when the input is negated:

Odd Function Test

f(-x) = -f(x) \quad \text{for all } x \in \text{Dom}(f)

Learn more about this formula: Odd Function Test →

Browse all functions formulas →

The graph is symmetric about the origin — rotating it

180°

around the origin produces the same curve.

The function

f(x) = x^3

is odd:

f(-x) = (-x)^3 = -x^3 = -f(x)

. So are

f(x) = x

,

f(x) = x^5

, and

f(x) = \sin(x)

.

To test algebraically: substitute

-x

for

x

, simplify, and compare to

f(x)

and

-f(x)

. If the result equals

f(x)

, the function is even. If it equals

-f(x)

, the function is odd. If neither, the function is neither even nor odd.

Most functions fall into the third category. The function

f(x) = x^2 + x

satisfies

f(-x) = x^2 - x

, which equals neither

f(x)

nor

-f(x)

.

For a function to be even or odd, its domain must be symmetric about the origin — if

x

is in the domain, so is

-x

.

The table below collects the three possible outcomes with the algebraic test, the graphical symmetry it implies, and a few examples for each.

Category	Algebraic test on f(−x)	Graphical symmetry	Examples
Even	f(−x) = f(x)	mirror symmetry about the y-axis	x², x⁴, \|x\|, cos(x)
Odd	f(−x) = −f(x)	point symmetry about the origin	x, x³, x⁵, sin(x)
Neither	result matches neither f(x) nor −f(x)	no symmetry of either type	x² + x, eˣ, ln(x)

f(x) = x³: symmetric about the origin

Rotating the picture half a turn about the origin returns it to itself: the point (x, y) has the partner (−x, −y) on the curve, which is f(−x) = −f(x). That is an odd function; the parabola beside it in the tool is the even case instead. Test any family for both symmetries on the symmetry explorer.

Most functions are neither even nor odd; the two symmetries are special properties, not a dichotomy.

Symmetry

Symmetry describes when one part of a graph determines another through reflection or rotation.

A graph symmetric about the

y

-axis has the property that the point

(x, y)

lies on the graph if and only if

(-x, y)

does. Every point on the right has a mirror image on the left. This is the graphical signature of even functions.

A graph symmetric about the origin has the property that

(x, y)

lies on the graph if and only if

(-x, -y)

does. Every point has a counterpart obtained by rotating

180°

around the origin. This is the graphical signature of odd functions.

A graph symmetric about the

x

-axis would have

(x, y)

on the graph whenever

(x, -y)

is. But such a curve fails the vertical line test — it cannot represent a function, since one

x

-value would correspond to two

y

-values.

Symmetry about other vertical lines is possible. The parabola

f(x) = (x - 3)^2

is symmetric about the line

x = 3

. Its vertex lies on this axis of symmetry, and the graph is a mirror image on either side.

Recognizing symmetry simplifies graphing. For an even function, plotting the right half determines the left. For an odd function, plotting one portion determines the symmetric portion.

f(x) = x²: symmetric about the y-axis

The reflection of the parabola across the y-axis lands exactly on the parabola: f(−x) = f(x). A graph symmetric about the origin, like x³, or symmetric about neither, like x² + x, can be checked the same way. Fold any curve along the axis or spin it about the origin on the symmetry explorer.

Symmetry about a vertical line other than the y-axis is a shifted even function.

One-to-One Functions

A function is one-to-one (or injective) if different inputs always produce different outputs. No two

x

-values share the same

y

-value. Equivalently, if

f(a) = f(b)

, then

a = b

.

The function

f(x) = 2x + 5

is one-to-one. If

2a + 5 = 2b + 5

, then

a = b

. Every output comes from exactly one input.

The function

f(x) = x^2

is not one-to-one on

(-\infty, \infty)

. Both

f(3) = 9

and

f(-3) = 9

— two different inputs produce the same output.

The horizontal line test identifies one-to-one functions graphically. If every horizontal line intersects the graph at most once, the function is one-to-one. If any horizontal line crosses twice or more, it is not.

One-to-one functions are precisely those that have inverse functions. Because each output comes from a unique input, the process can be reversed — given an output, identify the input that produced it. This reversal defines the inverse.

A function that is not one-to-one can become one-to-one by restricting its domain. The function

f(x) = x^2

restricted to

[0, \infty)

is one-to-one: positive inputs produce distinct outputs. This restriction enables defining

\sqrt{x}

as its inverse.

f(x) = x³: every horizontal line meets it once

The cubic passes the horizontal line test, so it is one-to-one and its inverse, the cube root, exists and is drawn as the mirror image in y = x. A parabola would fail the test, two inputs sharing one output, and would need its domain cut before an inverse could be drawn. Compare the two on the inverse function explorer.

One-to-one is exactly the property an inverse needs.

Onto Functions

A function is onto (or surjective) if every element of the codomain is actually achieved as an output. The range equals the codomain — nothing in the target set is missed.

Whether a function is onto depends on how the codomain is defined. The function

f(x) = x^2

from

\mathbb{R}

to

\mathbb{R}

is not onto: negative numbers are in the codomain

\mathbb{R}

but never appear as outputs. The same function from

\mathbb{R}

to

[0, \infty)

is onto: every non-negative number is the square of some real number.

The horizontal line test has a variant for onto: every horizontal line at a

y

-value in the codomain must intersect the graph at least once.

The function

f(x) = e^x

from

\mathbb{R}

to

\mathbb{R}

is not onto — outputs are always positive. From

\mathbb{R}

to

(0, \infty)

, it is onto — every positive number is

e^x

for some

x

.

The function

f(x) = x^3

from

\mathbb{R}

to

\mathbb{R}

is onto. Every real number is the cube of some real number.

In elementary algebra, codomain is often implicitly taken to be all real numbers, and the focus falls on range rather than the onto property. The distinction becomes important in abstract mathematics and formal definitions.

f(x) = eˆx: the outputs never reach 0 or below

The band on the y-axis marks the heights the curve reaches: everything above 0, with an open dot at 0 itself, and nothing at or below it. As a map from ℝ to ℝ, eˆx is therefore not onto: the negative half of the codomain is never hit. Shrink the codomain to (0, ∞) and the same function is onto. Compare the ranges of other families on the range explorer.

Whether a function is onto is a statement about the codomain as much as about the function.

Bijective Functions

A function is bijective if it is both one-to-one and onto. Every element of the codomain is achieved exactly once. There is a perfect pairing between domain and codomain — each input corresponds to exactly one output, and each output corresponds to exactly one input.

The function

f(x) = 3x - 7

from

\mathbb{R}

to

\mathbb{R}

is bijective. It is one-to-one: different inputs give different outputs. It is onto: every real number is

3x - 7

for some

x

.

The function

f(x) = x^2

from

\mathbb{R}

to

\mathbb{R}

is neither one-to-one nor onto, hence not bijective.

The function

f(x) = x^2

from

[0, \infty)

to

[0, \infty)

is bijective. Restricting the domain made it one-to-one; restricting the codomain to match the range made it onto.

Bijective functions have inverses that are also bijective. The inverse reverses the pairing: if

f(a) = b

, then

f^{-1}(b) = a

. Both functions establish the same one-to-one correspondence, just read in opposite directions.

Bijections are fundamental in counting arguments. Two sets have the same size if and only if there exists a bijection between them.

The table below sets the three related properties — one-to-one, onto, and bijective — side by side, each compared against the others by definition, graphical test, and a typical example.

Property	Definition	Graphical test	Example (from ℝ to ℝ)
One-to-one (injective)	distinct inputs always give distinct outputs	horizontal line test — every horizontal line meets the graph at most once	f(x) = 2x + 5
Onto (surjective)	every element of the codomain is achieved as an output	every horizontal line at a codomain height meets the graph at least once	f(x) = x³
Bijective	both injective and surjective — a perfect pairing	every horizontal line at a codomain height meets the graph exactly once	f(x) = 3x − 7

Increasing and Decreasing

A function is increasing on an interval if larger inputs produce larger outputs throughout that interval. For any

a

and

b

in the interval with

a < b

, we have

f(a) < f(b)

. The graph rises from left to right.

A function is decreasing on an interval if larger inputs produce smaller outputs. For

a < b

, we have

f(a) > f(b)

. The graph falls from left to right.

Strictly increasing and strictly decreasing emphasize that the inequality is strict — equal outputs are not allowed. Non-decreasing means

f(a) \leq f(b)

when

a < b

, permitting flat segments. Non-increasing means

f(a) \geq f(b)

, also permitting flat segments.

The function

f(x) = x^3

is strictly increasing on

(-\infty, \infty)

. The function

f(x) = -x

is strictly decreasing everywhere.

The function

f(x) = x^2

is decreasing on

(-\infty, 0]

and increasing on

[0, \infty)

. It switches direction at the vertex.

Increasing and decreasing intervals are expressed using

x

-values, describing portions of the domain. The statement "increasing on

(2, 5)

" means that within this interval, the graph rises.

Determining these intervals from a graph is visual: trace the curve and note where it rises, where it falls. From an equation, calculus provides systematic methods; in algebra, analyzing the structure of the formula often suffices.

Falling, then rising

The section's example f(x) = x². Read from left to right, the curve falls until the vertex at x = 0 and rises after it. The direction is stated as x-intervals under the plot: decreasing on (−∞, 0], increasing on [0, ∞).

Increasing and decreasing always describe stretches of the x-axis, never of the y-axis.

Monotonic Functions

A function is monotonic if it moves in only one direction — always increasing or always decreasing — across its entire domain. It never reverses course.

A monotonically increasing function satisfies

a < b \implies f(a) \leq f(b)

for all

a, b

in the domain. A strictly monotonically increasing function satisfies

a < b \implies f(a) < f(b)

.

A monotonically decreasing function satisfies

a < b \implies f(a) \geq f(b)

. A strictly monotonically decreasing function satisfies

a < b \implies f(a) > f(b)

.

Examples of monotonic functions:

f(x) = e^x

(strictly increasing),

f(x) = \ln(x)

(strictly increasing on its domain),

f(x) = -x^3

(strictly decreasing),

f(x) = 3

(constant, hence both non-decreasing and non-increasing).

Monotonic functions are always one-to-one (unless constant). Because they never produce the same output twice (in the strict case), they have inverses. This is why exponential functions have logarithmic inverses — both are strictly monotonic.

Many functions are monotonic on pieces of their domain but not overall. The function

f(x) = \sin(x)

increases on some intervals and decreases on others. It is not monotonic on

\mathbb{R}

, but it is strictly increasing on

[-\pi/2, \pi/2]

, enabling the definition of

\arcsin

on that restricted domain.

eˆx and ln x: both only ever rise

Neither curve turns back anywhere: eˆx rises across all real inputs and ln x across all positive ones, so both are strictly increasing, hence monotonic. That is exactly why each can be undone, and the two are mirror images in the dashed line y = x: each is the other’s inverse. Explore other inverse pairs on the inverse function explorer.

A strictly monotonic function never repeats an output, so it is always one-to-one.

Boundedness

A function is bounded above if its outputs never exceed some finite value. There exists a number

M

such that

f(x) \leq M

for all

x

in the domain. The graph never rises above the horizontal line

y = M

.

A function is bounded below if its outputs are never less than some finite value. There exists a number

m

such that

f(x) \geq m

for all

x

in the domain. The graph never falls below

y = m

.

A function is bounded if it is bounded both above and below — its outputs stay within a finite strip.

The function

f(x) = \sin(x)

is bounded:

-1 \leq \sin(x) \leq 1

for all

x

. The function

f(x) = x^2

is bounded below (by

0

) but not above — it grows without limit. The function

f(x) = x

is unbounded in both directions.

The function

f(x) = \dfrac{1}{1 + x^2}

is bounded:

0 < f(x) \leq 1

for all

x

.

Boundedness constrains the range. A function bounded above has range with a finite supremum; a function bounded below has range with a finite infimum. A bounded function has range contained in some finite interval.

f(x) = sin x: every output between −1 and 1

However far the wave runs in either direction, its heights stay inside the band from −1 to 1 on the y-axis, both ends reached and marked with closed dots. So sin x is bounded above by 1 and below by −1; the band is its whole range. See which families are bounded on the range explorer.

A bounded function's whole graph fits inside one horizontal strip.

Local Extrema

A local maximum occurs at

x = c

if

f(c) \geq f(x)

for all

x

near

c

. The function reaches a peak at

c

relative to its immediate neighborhood — it may be exceeded elsewhere, but not nearby.

A local minimum occurs at

x = c

if

f(c) \leq f(x)

for all

x

near

c

. The function reaches a valley at

c

relative to its immediate neighborhood.

Together, local maxima and local minima are called local extrema.

On a graph, local maxima appear as peaks — points where the curve rises and then falls. Local minima appear as valleys — points where the curve falls and then rises.

The function

f(x) = x^3 - 3x

has a local maximum at

x = -1

(where

f(-1) = 2

) and a local minimum at

x = 1

(where

f(1) = -2

). Between these points, the function decreases.

Local extrema occur at turning points — places where the function switches between increasing and decreasing. They can also occur at endpoints of a closed domain, where the function might achieve a high or low relative to the interior.

Finding local extrema algebraically is a central problem in calculus. In algebra, analyzing the structure of specific function types — vertices of parabolas, critical points of cubics — provides the answer.

A cubic’s peak and valley

The curve 0.2x³ − 2x rises to a peak, falls through the origin into a valley, then rises for good. The peak is a local maximum and the valley a local minimum: each is extreme only among nearby points, since the curve climbs higher than the peak further right and falls lower than the valley further left. Compare the shapes of other families on the function types explorer.

Local extrema sit exactly where the function changes between increasing and decreasing.

Absolute Extrema

The absolute maximum (or global maximum) of a function is the largest value it achieves anywhere on its domain. If

f(c)

is the absolute maximum, then

f(c) \geq f(x)

for every

x

in the domain.

The absolute minimum (or global minimum) is the smallest value achieved anywhere on the domain. If

f(c)

is the absolute minimum, then

f(c) \leq f(x)

for every

x

in the domain.

Absolute extrema are the overall champions — the highest peak and lowest valley across the entire function, not just in a neighborhood.

Not every function has absolute extrema. The function

f(x) = x

on

(-\infty, \infty)

has no maximum or minimum — it extends without bound in both directions. The function

f(x) = 1/x

on

(0, 1)

approaches infinity as

x \to 0^+

and approaches

1

as

x \to 1^-

, but achieves neither limit.

On a closed, bounded interval, a continuous function always achieves both an absolute maximum and an absolute minimum. This is the Extreme Value Theorem. The extrema occur either at local extrema within the interval or at the endpoints.

To find absolute extrema on

[a, b]

: identify all local extrema in

(a, b)

, evaluate the function at these points and at the endpoints

a

and

b

, and compare to find the largest and smallest values.

The table below contrasts the two notions of extremum — local relative to a neighborhood, absolute relative to the entire domain — and shows when each is guaranteed to exist.

Type	Definition	Scope	Existence guarantee
Local max / min	f(c) is the largest / smallest value among nearby points	a neighborhood of c	none in general — depends on the function
Absolute (global) max / min	f(c) is the largest / smallest value the function ever takes	the entire domain	guaranteed for continuous f on a closed, bounded interval (Extreme Value Theorem)

Continuity

A function is continuous at a point if its graph has no break there — no jump, no hole, no explosion to infinity. Intuitively, the graph can be drawn through that point without lifting the pencil.

Formally,

f

is continuous at

x = c

if three conditions hold:

f(c)

is defined, the limit of

f(x)

as

x \to c

exists, and the limit equals

f(c)

. The output at

c

matches what the nearby outputs predict.

A function is continuous on an interval if it is continuous at every point in that interval. A function is continuous everywhere if it is continuous on its entire domain.

Polynomials are continuous everywhere. Rational functions are continuous wherever defined — at every point except where the denominator is zero. Exponential, logarithmic, and trigonometric functions are continuous on their domains.

Continuity guarantees certain behaviors. A continuous function on a closed interval achieves its maximum and minimum values. A continuous function that is negative at one point and positive at another must cross zero somewhere in between (Intermediate Value Theorem).

Most functions encountered in elementary mathematics are continuous on their domains. Discontinuities, when they occur, are special events worth noting and classifying.

Types of Discontinuities

When continuity fails, the type of failure is classified by what goes wrong.

A removable discontinuity (or hole) occurs when the limit exists but the function is either undefined or has a different value at that point. The graph has a single missing point — a hole that could be "filled in" by redefining the function there. The function

f(x) = \dfrac{x^2 - 1}{x - 1}

has a removable discontinuity at

x = 1

.

A jump discontinuity occurs when the left-hand limit and right-hand limit both exist but differ. The function jumps from one value to another. Piecewise functions often have jump discontinuities at the boundaries between pieces.

An infinite discontinuity occurs when the function approaches infinity (or negative infinity) as

x

approaches some value. This manifests as a vertical asymptote. The function

f(x) = 1/x

has an infinite discontinuity at

x = 0

.

An oscillating discontinuity occurs when the function oscillates infinitely often near a point without settling to any limit. The function

f(x) = \sin(1/x)

near

x = 0

is the classic example.

Identifying the type of discontinuity guides how to handle it — removable discontinuities can be eliminated by redefining the function; others cannot. The table below collects the four standard discontinuity types with the behavior that defines each.

Type	What happens at the point	Recoverable?	Example
Removable (hole)	limit exists but the function is undefined or differently valued there	yes — redefine f at the point	(x² − 1) ⁄ (x − 1) at x = 1
Jump	left and right limits both exist but differ	no — the gap is structural	step or piecewise functions at a boundary
Infinite	function diverges to ±∞ near the point	no — produces a vertical asymptote	1 ⁄ x at x = 0
Oscillating	function oscillates infinitely without settling to any limit	no — no limit exists	sin(1 ⁄ x) near x = 0

Asymptotes

An asymptote is a line that a graph approaches but typically never reaches. Asymptotes describe limiting behavior — what happens as the function extends toward infinity or toward an undefined point.

A vertical asymptote is a vertical line

x = a

that the graph approaches as outputs explode toward

\pm\infty

. The function

f(x) = 1/(x - 2)

has a vertical asymptote at

x = 2

. As

x \to 2

, the outputs grow without bound.

A horizontal asymptote is a horizontal line

y = L

that the graph approaches as

x \to \infty

or

x \to -\infty

. The function

f(x) = 1/x

has horizontal asymptote

y = 0

. As

x

grows large, the outputs approach zero.

An oblique (or slant) asymptote is a non-horizontal line

y = mx + b

that the graph approaches as

x \to \pm\infty

. This occurs in rational functions when the numerator's degree exceeds the denominator's degree by exactly one. The function

f(x) = \dfrac{x^2 + 1}{x}

has oblique asymptote

y = x

.

A function can cross a horizontal or oblique asymptote; it describes limiting behavior, not a barrier. A function never crosses a vertical asymptote — the function is undefined there.

Asymptotes shape the large-scale structure of a graph, providing guidelines the curve follows at its extremes. The table below collects the three asymptote types with the form of the line, the condition that produces them, and a worked example.

Type	Form of the line	When it occurs	Example
Vertical	x = a	f(x) → ±∞ as x → a; cannot be crossed	f(x) = 1 ⁄ (x − 2) at x = 2
Horizontal	y = L	f(x) → L as x → ±∞; may be crossed	f(x) = 1 ⁄ x has y = 0
Oblique (slant)	y = mx + b	rational with numerator degree exactly one above denominator	f(x) = (x² + 1) ⁄ x has y = x

f(x) = 1/x: vertical x = 0 and horizontal y = 0

The two dashed lines are the asymptotes: the curve runs off to infinity beside the vertical one and flattens onto the horizontal one at both ends, approaching each without touching. The vertical line marks a forbidden input, the horizontal one a limiting output. Explore oblique asymptotes as well on the asymptotes explorer.

Asymptotes are statements about limits drawn as lines.

End Behavior

End behavior describes how

f(x)

behaves as

x

moves toward positive or negative infinity — the far right and far left of the graph.

The notation

f(x) \to L

as

x \to \infty

means the outputs approach the value

L

as

x

grows without bound. If

L

is finite, this indicates a horizontal asymptote.

The notation

f(x) \to \infty

as

x \to \infty

means the outputs grow without bound — the graph rises indefinitely on the right.

For polynomials, end behavior is determined by the leading term. The function

f(x) = 3x^4 - 5x^2 + 2

behaves like

3x^4

for large

|x|

: rising on both ends since the degree is even and the leading coefficient is positive.

For rational functions, compare the degrees of numerator and denominator. If the numerator has lower degree,

f(x) \to 0

. If degrees are equal,

f(x)

approaches the ratio of leading coefficients. If the numerator has higher degree,

f(x)

is unbounded.

For exponential functions,

a^x \to \infty

as

x \to \infty

when

a > 1

, and

a^x \to 0

as

x \to \infty

when

0 < a < 1

.

End behavior provides the frame for the entire graph — knowing where the curve ultimately goes constrains what can happen in between.

The leading term decides the ends

The section's polynomial 3x⁴ − 5x² + 2 (solid) next to its leading term 3x⁴ (dashed). Near the origin the two differ, but as |x| grows the lower-degree terms stop mattering and the curves run together: even degree with a positive leading coefficient, so both ends rise without bound.

For any polynomial, the degree and the sign of the leading coefficient settle the end behavior.

Periodicity

A function is periodic if it repeats its values at regular intervals. There exists a positive number

p

such that

f(x + p) = f(x)

for all

x

in the domain. The graph shows the same pattern cycling endlessly.

The smallest such

p

is called the fundamental period (or simply the period). The function

f(x) = \sin(x)

has period

2\pi

: the wave from

0

to

2\pi

repeats identically thereafter.

Periodic functions have bounded range — the repeating pattern confines outputs to a fixed interval. The sine and cosine functions have range

[-1, 1]

.

The amplitude measures half the vertical distance between maximum and minimum values. For

f(x) = \sin(x)

, the amplitude is

1

. For

f(x) = 3\sin(x)

, the amplitude is

3

.

Periodic functions model cyclic phenomena: sound waves, tides, seasons, heartbeats, alternating current. The period corresponds to the duration of one cycle; the amplitude corresponds to the intensity of oscillation.

Not all functions are periodic. Polynomials, exponentials, and logarithms never repeat exactly. Linear combinations of periodic functions with incommensurable periods can produce non-periodic behavior.

f(x) = sin x repeats every 2π

Any stretch of the curve of length 2π is a copy of every other such stretch: that is periodicity, and 2π is the smallest length that works. The values never leave [−1, 1], so the function is also bounded. Compare with the non-periodic families on the function types explorer.

Periodicity is why one cycle of a trigonometric function is enough to know all of it.

Zeros and Roots

A zero of a function is an input value where the output equals zero:

f(c) = 0

. Zeros are also called roots of the function.

Graphically, zeros are

x

-intercepts — points where the curve crosses or touches the horizontal axis.

Finding zeros means solving the equation

f(x) = 0

. For

f(x) = x^2 - 5x + 6

, the zeros are the solutions to

x^2 - 5x + 6 = 0

. Factoring gives

(x - 2)(x - 3) = 0

, so the zeros are

x = 2

and

x = 3

.

The multiplicity of a zero refers to how many times the corresponding factor appears. If

f(x) = (x - 2)^3(x + 1)

, then

x = 2

is a zero of multiplicity

3

and

x = -1

is a zero of multiplicity

1

.

Multiplicity affects graph behavior. A zero of odd multiplicity crosses the axis — the graph passes through. A zero of even multiplicity touches the axis and turns back — the graph bounces off.

Zeros are fundamental in equation solving, factoring, and sign analysis. Knowing where the function equals zero divides the domain into intervals where the function is positive or negative.

Crossing and touching zeros

f(x) = (x + 1)(x − 2)² has two zeros. The factor x + 1 appears once, so at x = −1 the graph passes through the axis and changes sign. The factor x − 2 appears twice, so at x = 2 the graph touches the axis and turns back without changing sign.

Reading the multiplicities tells you how the graph meets the axis before you plot a single point.

Average Rate of Change

The average rate of change of a function over an interval

[a, b]

measures how much the output changes per unit of input:

Average Rate of Change

\frac{f(b) - f(a)}{b - a}

Learn more about this formula: Average Rate of Change →

Browse all functions formulas →

This is the slope of the secant line connecting the points

(a, f(a))

and

(b, f(b))

on the graph.

For

f(x) = x^2

on

[1, 3]

:

\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4

The function increases by an average of

4

units in output for each unit of input over this interval.

Average rate of change has units: output units per input unit. If

f(t)

gives position in meters after

t

seconds, the average rate of change is average velocity in meters per second.

A positive average rate indicates overall increase over the interval. A negative rate indicates overall decrease. Zero means the function returns to its starting value — though it may have varied in between.

Average rate of change over shrinking intervals approaches the instantaneous rate of change — the derivative — which is the focus of calculus. In algebra, the average rate provides a coarse measure of how fast a function changes.

Average rate of change as a slope

The section's example: f(x) = x² on [1, 3]. The output climbs from 1 to 9, a rise of 8, while the input moves from 1 to 3, a run of 2. The ratio 8 / 2 = 4 is the slope of the secant line joining (1, 1) and (3, 9), the amber line.

Shrinking the interval turns the secant into the tangent, and the average rate into the instantaneous one.

Concavity

Concavity describes how a curve bends — whether it curves upward like a bowl or downward like an arch.

A function is concave up on an interval if its graph bends upward: it lies above its tangent lines, and the rate of change is increasing. A concave-up curve holds water if imagined as a bowl.

A function is concave down on an interval if its graph bends downward: it lies below its tangent lines, and the rate of change is decreasing. A concave-down curve spills water.

An inflection point is where concavity changes — where the curve switches from bending up to bending down, or vice versa. The cubic

f(x) = x^3

has an inflection point at the origin: concave down for

x < 0

, concave up for

x > 0

.

Visually, concavity is detected by the curve's shape. A parabola opening upward is entirely concave up. A parabola opening downward is entirely concave down. Cubics and higher-degree polynomials may have regions of both types.

Concavity affects how well a linear approximation fits a curve. Concave-up regions curve away from tangent lines in one direction; concave-down regions curve away in the other.

Full analysis of concavity uses calculus — specifically, the second derivative. In algebra, recognizing concavity provides qualitative understanding of graph shape.

Bending down, then bending up

The section's cubic f(x) = x³. Left of the origin the curve lies below its tangent line (dashed, drawn at x = −1): concave down. Right of the origin it lies above its tangent (dashed, at x = 1): concave up. The switch happens at the origin, the inflection point.

The same comparison with tangent lines works for any curve, wherever it bends.

Properties at a Glance

The properties covered on this page describe distinct facets of a function's character — its symmetry, its monotonicity, its boundaries, its limits, its turning points, and the shape of its bending. The table below collects each property with a one-line summary of what it means and the visual signature that reveals it on a graph — useful as a reference card or as a checklist when analyzing an unfamiliar function.

Property	What it means	Visual signature on the graph
Even	f(−x) = f(x) for every x in the domain	mirror symmetry about the y-axis
Odd	f(−x) = −f(x) for every x in the domain	point symmetry about the origin
One-to-one	distinct inputs always give distinct outputs	passes the horizontal line test
Onto	range covers the entire codomain	every horizontal line at a codomain height meets the graph
Bijective	both one-to-one and onto	every horizontal line at a codomain height meets the graph exactly once
Increasing on (a, b)	larger inputs yield larger outputs on that interval	curve rises from left to right on (a, b)
Decreasing on (a, b)	larger inputs yield smaller outputs on that interval	curve falls from left to right on (a, b)
Monotonic	moves in only one direction over the whole domain	curve never reverses — always rising or always falling
Bounded	outputs stay within a fixed interval [m, M]	graph confined between two horizontal lines
Local extremum	a peak or valley relative to nearby points	turning point where curve switches direction
Absolute extremum	the overall highest or lowest output on the entire domain	the highest peak or lowest valley anywhere on the curve
Continuous	no breaks anywhere on the domain	curve can be drawn without lifting the pencil
Has an asymptote	approaches a line at extremes or near a singular point	curve gets arbitrarily close to a vertical, horizontal, or slanted line
Periodic	f(x + p) = f(x) for some smallest period p	a single pattern repeats unchanged every p units
Has a zero at c	f(c) = 0	curve crosses or touches the x-axis at x = c
Concave up	rate of change is increasing on the interval	curve bends upward — bowl-shaped
Concave down	rate of change is decreasing on the interval	curve bends downward — arch-shaped

Function Properties FAQ

How do you determine if a function is even or odd?

+

Substitute −x into the formula. Getting the original function back means it is even, and getting the exact negative of it means it is odd. Anything else makes it neither, which is the most common outcome. Even functions are symmetric about the vertical axis, odd ones about the origin.Read more →

What is a one-to-one function and why does it matter?

+

It is a function that never produces the same output twice, so distinct inputs always give distinct outputs. It matters because only one-to-one functions can be inverted: if two inputs shared an output, the reverse assignment would not know which to return. The horizontal line test checks the condition graphically.Read more →

Can a function be continuous but not differentiable?

+

Yes. Continuity only requires the graph to have no breaks, while differentiability additionally requires no sharp corners. The absolute value function is continuous everywhere but has a corner at zero, where no single tangent direction exists. Differentiability implies continuity, but the implication does not run the other way.Read more →

What is the difference between a removable and a non-removable discontinuity?

+

A removable discontinuity is a single missing or misplaced point that could be patched by redefining the function there, showing as a hole. Non-removable ones cannot be fixed by changing one value: a jump has the two sides landing at different heights, and an infinite discontinuity has the function growing without bound.Read more →

Can a function have more than one horizontal asymptote?

+

Yes, at most two, one for each direction. The behaviour as the input grows large positively can differ from the behaviour as it grows large negatively. Rational functions have the same asymptote both ways, but functions involving absolute values or exponentials can approach different heights on the two sides.Read more →

The Character of Functions

Key Terms

Symmetry & Type

Behavior

Structure

Even and Odd Functions

Symmetry

One-to-One Functions

Onto Functions

Bijective Functions

Increasing and Decreasing

Monotonic Functions

Boundedness

Local Extrema

Absolute Extrema

Continuity

Types of Discontinuities

Asymptotes

End Behavior

Periodicity

Zeros and Roots

Average Rate of Change

Concavity

Properties at a Glance

Function Properties FAQ

How do you determine if a function is even or odd?

What is a one-to-one function and why does it matter?

Can a function be continuous but not differentiable?

What is the difference between a removable and a non-removable discontinuity?

Can a function have more than one horizontal asymptote?