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Radical Functions



Key Terms
The Square Root Function
Graph of the Square Root Function
The Cube Root Function
Graph of the Cube Root Function
General Nth Root Functions
Domain of Radical Functions
Transformations
Inverse Relationship with Power Functions
Finding Inverses of Radical Functions
Graphing Strategy
Parent Functions at a Glance
Radical Functions FAQ



Radicals as Functions

When the radicand contains a variable, the radical becomes a function. Each input produces an output through the root operation. The index determines which inputs are valid and what shape the graph takes.

Square root functions and cube root functions are the most common, but the patterns extend to any index. Understanding these functions connects radicals to the broader study of function behavior.

Key Terms

Function Concepts

Radical Function— a function of the form f(x)=g(x)nf(x) = \sqrt[n]{g(x)}, where the radicand contains a variable
Square Root— the parent function f(x)=xf(x) = \sqrt{x} with domain [0,∞)[0, \infty)
Cube Root— the parent function f(x)=x3f(x) = \sqrt[3]{x} with domain (−∞,∞)(-\infty, \infty)

Determining Behavior

Index— even index restricts domain to non-negative radicands; odd index allows all reals
Principal Root— ensures the radical produces one output per input, making it a function
Rational Exponent— connects radical functions to power functions as inverses

See All Algebra Definitions →


The Square Root Function

The parent square root function is:

f(x)=xf(x) = \sqrt{x}


For each non-negative input, it returns the principal square root.

Domain: [0,∞)[0, \infty) — only non-negative inputs allowed, following the properties of radicals.

Range: [0,∞)[0, \infty) — outputs are non-negative by the principal root convention.

Key points on the graph:

(0,0),(1,1),(4,2),(9,3),(16,4)(0, 0), \quad (1, 1), \quad (4, 2), \quad (9, 3), \quad (16, 4)


The graph starts at the origin and rises to the right. It increases throughout its domain but the rate of increase slows — the curve bends downward, concave down.

The function is one-to-one. It passes the horizontal line test and has an inverse.

Graph of the Square Root Function

The square root graph has a distinctive shape: it begins at a point and curves gradually upward to the right.

At x=0x = 0, the graph has an endpoint — no values exist to the left.

As xx increases, x\sqrt{x} increases, but each additional unit of input produces less additional output. From x=0x = 0 to x=1x = 1, the output increases by 1. From x=1x = 1 to x=4x = 4, only another 1. From x=4x = 4 to x=9x = 9, another 1.

100=10,10000=100\sqrt{100} = 10, \quad \sqrt{10000} = 100


Multiplying the input by 100 only multiplies the output by 10.

The curve is the upper half of a parabola lying on its side. This connection becomes clear when considering inverse functions — the square root is the inverse of squaring restricted to non-negative values.
sqrt x rises by 1 over runs of 1, 3 and 5 in xxy12345678910−11234run 3run 5y = √x(1, 1)(4, 2)(9, 3)each +1 in output needs a longer run: 1, 3, 5Equal rises in √x need ever longer runs in x: 1, 3, 5, …
Equal rises, longer runs

The section’s steps on the graph of y = √x. The output climbs from 1 to 2 while x runs from 1 to 4, a run of 3, and from 2 to 3 while x runs from 4 to 9, a run of 5. Each further unit of height costs a longer run: the curve keeps rising but flattens.

The runs grow 1, 3, 5, …: the odd numbers, since (n + 1)² − n² = 2n + 1.

The Cube Root Function

The parent cube root function is:

f(x)=x3f(x) = \sqrt[3]{x}


Unlike the square root, it accepts all real inputs.

Domain: (−∞,∞)(-\infty, \infty) — any real number has a real cube root.

Range: (−∞,∞)(-\infty, \infty) — outputs span all real numbers.

Key points:

(−8,−2),(−1,−1),(0,0),(1,1),(8,2)(-8, -2), \quad (-1, -1), \quad (0, 0), \quad (1, 1), \quad (8, 2)


The graph passes through the origin and extends in both directions. Negative inputs yield negative outputs; positive inputs yield positive outputs. The function preserves sign, consistent with properties of odd-index radicals.

The cube root function is also one-to-one, with the cubing function as its inverse.

Graph of the Cube Root Function

The cube root graph has an S-like shape, though gentler than a cubic.

At x=0x = 0, the graph passes through the origin — not an endpoint but an inflection point where concavity changes.

For x>0x > 0, the curve rises but flattens as xx increases. For x<0x < 0, it falls but flattens as xx decreases toward negative infinity.

The function is symmetric about the origin:

f(−x)=−f(x)f(-x) = -f(x)


This is odd symmetry. Reflecting the graph across both axes leaves it unchanged.

Growth is slow. Large inputs produce relatively modest outputs:

10003=10,10000003=100\sqrt[3]{1000} = 10, \quad \sqrt[3]{1000000} = 100


Cube roots tame large numbers, compressing wide ranges into narrower ones.
The cube root graph is symmetric about the origin: (8, 2) and (-8, -2)xy−10−8−6−4−2246810−3−2−1123y = ∛x(8, 2)(−8, −2)f(−x) = −f(x)The cube root is odd: ∛(−8) = −2 mirrors ∛8 = 2 through O
Odd symmetry of the cube root

∛8 = 2 and ∛(−8) = −2, so (8, 2) and (−8, −2) sit on the curve, joined by a dashed segment through the origin. Every point (a, b) has its partner (−a, −b): a half-turn about the origin carries the curve onto itself, which is f(−x) = −f(x).

The square root has no such partner points: its graph lives only at x ≥ 0.

General Nth Root Functions

The pattern extends to any positive integer index:

f(x)=xnf(x) = \sqrt[n]{x}


For even nn (fourth root, sixth root, etc.):

Domain: [0,∞)[0, \infty)

Range: [0,∞)[0, \infty)

Shape resembles the square root — starts at origin, rises to the right, concave down.

Higher even indices produce flatter curves. The fourth root rises more slowly than the square root.

For odd nn (fifth root, seventh root, etc.):

Domain: (−∞,∞)(-\infty, \infty)

Range: (−∞,∞)(-\infty, \infty)

Shape resembles the cube root — S-curve through origin, odd symmetry.

Higher odd indices produce flatter curves. The fifth root is flatter than the cube root.

All nnth root functions are one-to-one and serve as inverses to the corresponding power functions, with domain restrictions for even indices.
sqrt x and the fourth root of x: equal at 0 and 1, the fourth root lower for x > 1xy246810121416−112345y = √xy = ⁴√xboth 1 at x = 1√16 = 4⁴√16 = 2higher even index: flatter, still starting at the originBoth even roots need x ≥ 0; the fourth root rises more slowly
Square root and fourth root

Both curves start at the origin and need x ≥ 0. They meet again at x = 1, where both equal 1. Beyond that the fourth root stays lower: at x = 16 it reaches only 2 while √16 = 4. The higher even index gives the flatter curve.

Between 0 and 1 the order flips: there the fourth root is the higher curve.

Domain of Radical Functions

When the radicand is an expression, solve for valid inputs.

f(x)=x−3f(x) = \sqrt{x - 3}


The radicand x−3x - 3 must be non-negative:

x−3≥0⇒x≥3x - 3 \geq 0 \quad \Rightarrow \quad x \geq 3


Domain: [3,∞)[3, \infty).

g(x)=5−2xg(x) = \sqrt{5 - 2x}


5−2x≥0⇒x≤525 - 2x \geq 0 \quad \Rightarrow \quad x \leq \frac{5}{2}


Domain: (−∞,52](-\infty, \frac{5}{2}].

h(x)=x+43h(x) = \sqrt[3]{x + 4}


Odd index — no restriction on the radicand.

Domain: (−∞,∞)(-\infty, \infty).

These domain calculations use the same principles as simplifying expressions and solving radical equations. The index determines whether restrictions apply.
Domains: sqrt(x-3) for x >= 3, sqrt(5-2x) for x <= 5/2, cube root of (x+4) for all xxy−6−5−4−3−2−112345678−3−2−112345y = ∛(x+4): all xy = √(5−2x): x ≤ 5/2y = √(x−3): x ≥ 3even roots stop where the radicand reaches 0Even-index radicals end at radicand 0; the cube root runs on
Where each radical exists

The section’s three functions. √(x − 3) begins at x = 3 and runs right; √(5 − 2x) runs in from the left and ends at x = 5/2. Each even root stops exactly where its radicand reaches 0. ∛(x + 4) has no such point: it continues through x = −4 and across the whole axis.

For an even root, solve radicand ≥ 0; for an odd root, there is nothing to solve.

Transformations

Radical functions transform like any function family.

Vertical shift: f(x)=x+kf(x) = \sqrt{x} + k

Shifts the graph up if k>0k > 0, down if k<0k < 0. Range becomes [k,∞)[k, \infty).

Horizontal shift: f(x)=x−hf(x) = \sqrt{x - h}

Shifts the graph right if h>0h > 0, left if h<0h < 0. Domain becomes [h,∞)[h, \infty).

Vertical stretch or compression: f(x)=axf(x) = a\sqrt{x}

Stretches if ∣a∣>1|a| > 1, compresses if ∣a∣<1|a| < 1.

Reflection over the xx-axis: f(x)=−xf(x) = -\sqrt{x}

The graph flips upside down. Range becomes (−∞,0](-\infty, 0].

Reflection over the yy-axis: f(x)=−xf(x) = \sqrt{-x}

The graph reflects horizontally. Domain becomes (−∞,0](-\infty, 0].

Combined example:

f(x)=2x−1+3f(x) = 2\sqrt{x - 1} + 3


Start at (1,3)(1, 3), stretched vertically by factor 2, rising to the right.
Transformation Form Effect on graph Effect on domain / range
Vertical shift √x + k shifts up (k > 0) or down (k < 0) range becomes [k, ∞)
Horizontal shift √(x − h) shifts right (h > 0) or left (h < 0) domain becomes [h, ∞)
Vertical stretch / compression a √x stretches (|a| > 1) or compresses (|a| < 1) no change
Reflection over x-axis − √x flips upside down range becomes (−∞, 0]
Reflection over y-axis √(−x) flips horizontally domain becomes (−∞, 0]

Inverse Relationship with Power Functions

Radical functions and power functions are inverses.

The square root function f(x)=xf(x) = \sqrt{x} is the inverse of g(x)=x2g(x) = x^2 when gg is restricted to x≥0x \geq 0.

f(g(x))=x2=xfor x≥0f(g(x)) = \sqrt{x^2} = x \quad \text{for } x \geq 0


g(f(x))=(x)2=xfor x≥0g(f(x)) = (\sqrt{x})^2 = x \quad \text{for } x \geq 0


The cube root function f(x)=x3f(x) = \sqrt[3]{x} is the inverse of g(x)=x3g(x) = x^3 with no restriction needed.

f(g(x))=x33=xfor all real xf(g(x)) = \sqrt[3]{x^3} = x \quad \text{for all real } x


g(f(x))=(x3)3=xfor all real xg(f(x)) = (\sqrt[3]{x})^3 = x \quad \text{for all real } x


Graphically, inverse functions reflect across the line y=xy = x. The parabola y=x2y = x^2 and the square root curve are mirror images across this diagonal.

The identity x2=∣x∣\sqrt{x^2} = |x| from properties explains why the restriction x≥0x \geq 0 is needed for the square function. Without it, the square function is not one-to-one and has no inverse.

Finding Inverses of Radical Functions

To find the inverse of a radical function, swap xx and yy, then solve.

f(x)=x−5,x≥5f(x) = \sqrt{x - 5}, \quad x \geq 5


Write as y=x−5y = \sqrt{x - 5}.

Swap: x=y−5x = \sqrt{y - 5}

Solve: x2=y−5x^2 = y - 5, so y=x2+5y = x^2 + 5

The inverse is f−1(x)=x2+5f^{-1}(x) = x^2 + 5, with domain x≥0x \geq 0 (matching the range of the original).

g(x)=2x+13g(x) = \sqrt[3]{2x + 1}


Swap: x=2y+13x = \sqrt[3]{2y + 1}

Cube: x3=2y+1x^3 = 2y + 1

Solve: y=x3−12y = \frac{x^3 - 1}{2}

The inverse is g−1(x)=x3−12g^{-1}(x) = \frac{x^3 - 1}{2}, valid for all real xx.

The connection between rational exponents and roots makes these inverse relationships algebraically natural. Roots undo powers; powers undo roots.
f(x) = sqrt(x-5) and its inverse x^2 + 5, x >= 0, mirror images across y = xxy−1123456789101112−1123456789101112y = xf(x) = √(x − 5)f⁻¹(x) = x² + 5, x ≥ 0(9, 2)(2, 9)starts at (5, 0)starts at (0, 5)Swapping x and y: f⁻¹ takes inputs x ≥ 0, the outputs of f
A radical and its inverse

f(x) = √(x − 5) starts at (5, 0); its inverse x² + 5 starts at (0, 5). The point (9, 2) on f becomes (2, 9) on the inverse, mirrored across the dashed line y = x. The outputs of f are the numbers y ≥ 0, so the inverse accepts only x ≥ 0.

Without the restriction x ≥ 0, x² + 5 would not be one-to-one and could not be the inverse.

Graphing Strategy

To graph a transformed radical function:

Identify the parent function based on the index.

Determine the domain by setting the radicand appropriately — non-negative for even index, unrestricted for odd.

Find the starting point (for even index) or the center point (for odd index) after applying shifts.

Plot additional points using the transformed function.

Sketch the curve, respecting concavity — down for even index, changing at inflection for odd index.

Example: f(x)=−x+2+4f(x) = -\sqrt{x + 2} + 4

Parent: x\sqrt{x}

Domain: x+2≥0x + 2 \geq 0, so x≥−2x \geq -2

Starting point: (−2,4)(-2, 4)

Reflected over xx-axis, so the curve descends to the right.

Additional point: x=2x = 2 gives f(2)=−4+4=−2+4=2f(2) = -\sqrt{4} + 4 = -2 + 4 = 2

The graph starts at (−2,4)(-2, 4) and curves downward to the right.
Step Action What it gives you
1 identify the parent function based on the index square root family (even) or cube root family (odd)
2 determine the domain by setting the radicand appropriately valid input range
3 find the starting point (even index) or center point (odd index) after applying shifts anchor for the graph
4 plot additional points using the transformed function shape verification
5 sketch the curve, respecting concavity finished graph

Parent Functions at a Glance

The two parent radical functions covered in the sections above can be set side by side as a single reference. The table below collects their domains and ranges, key points, graph shape and symmetry, inverses, and how each pattern generalizes to higher-index roots — useful as a study card and as a quick check when working with transformations or finding inverses.
Property Square root   f(x) = √x Cube root   f(x) = ∛x
Domain [0, ∞) (−∞, ∞)
Range [0, ∞) (−∞, ∞)
Key points (0, 0), (1, 1), (4, 2), (9, 3) (−8, −2), (0, 0), (1, 1), (8, 2)
Graph shape & symmetry starts at origin, rises and bends, concave down; no symmetry S-curve through origin; odd symmetry, f(−x) = −f(x)
Inverse function x²   (restricted to x ≥ 0) x³   (no restriction)
Generalizes to even-index n√x — flatter for higher n odd-index n√x — flatter for higher n

Radical Functions FAQ

Why does the square root graph flatten out as xx grows?

+
Because the output grows far more slowly than the input. Going from x=0x = 0 to x=1x = 1 raises the output by 11; earning the next 11 takes you all the way out to x=4x = 4, and the one after that to x=9x = 9. Put sharply, 100=10\sqrt{100} = 10 but 10000=100\sqrt{10000} = 100, so multiplying the input by 100100 only multiplies the output by 1010.Read more →

Why does x−h\sqrt{x - h} shift the graph right rather than left?

+
Because the endpoint sits where the radicand is zero, and x−h=0x - h = 0 means x=hx = h. Subtracting hh moves the starting point out to hh, so a minus sign shifts right. The vertical case only looks opposite because +k+k acts on the output directly, lifting the whole curve. Inputs and outputs are adjusted on different sides of the function.Read more →

Why isn't −x\sqrt{-x} undefined?

+
Because −x-x is positive whenever xx is negative. The requirement is that the radicand be non-negative, not that the expression under the bar look positive. Solving −x≥0-x \geq 0 gives x≤0x \leq 0, so the domain is (−∞,0](-\infty, 0] and the graph is the square root curve reflected across the yy-axis.Read more →

Why does the inverse of x\sqrt{x} need the restriction x≥0x \geq 0?

+
Because x2x^2 on its own is not one-to-one: 33 and −3-3 both square to 99, so there is no single value to send 99 back to. Restricting the squaring function to x≥0x \geq 0 makes it one-to-one and gives x\sqrt{x} something to invert. Cube roots need no such restriction, since x3x^3 is already one-to-one over all reals.Read more →