Topics
7. Inverse Functions and Logarithms
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United States of America
Grades 9-12

7.02 Radical functions

Lesson
Practice

Introduction

In lesson  7.01 Inverse functions  , we identified and graphed the inverses of functions including radical, exponential, rational, and polynomial functions. This lesson looks specifically at the inverse of power functions.

Radical functions

Inverses of the power functions f \left(x \right) = x^n, with domain restrictions as needed, form another type of parent function, called radical functions, where f \left( x \right) = \sqrt[n]x.

Radical function

A function that includes a radical expression with the independent variable in the radicand

Radicand

The value or expression inside the radical symbol

The nth root of x. The variable x is highlighted.

The index of a radical function can be any real number but the most common are those containing square roots, called a square root function, and cube roots, called a cube root function.

Index (of a radical)

The number on a radical symbol that indicates which type of root it represents. The index on a square root is 2 and the index on a cube root is 3.

The nth root of x. The variable n is highlighted.
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Consider the graph of the square root function y=\sqrt{x} and its key features:

  • Domain: \left[0, \infty \right)

  • Range: \left[0, \infty \right)

  • x-intercept: \left(0,0\right)
  • y-intercept: \left(0,0\right)

  • Increasing over its domain

  • As x \to \infty, y \to \infty

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Consider the graph of. the cube root function {y=\sqrt[3]{x}} and its key features:

  • Domain: \left(-\infty, \infty \right)

  • Range: \left(-\infty, \infty \right)

  • x-intercept: \left(0,0\right)
  • y-intercept: \left(0,0\right)

  • Point of inflection: \left(0,0\right)

  • Increasing over its domain

  • As x \to \infty, y \to \infty

  • As x \to -\infty, y \to -\infty
  • Odd function

Radical functions can be transformed similarly to any transformation of the parent function, y= af \left(x-h \right) +k, discussed in lesson  1.03 Function transformations  :

Square rootCube root
\text{Parent function:}y= \sqrt{x}y= \sqrt[3]{x}
\text{Reflection across the }x\text{-axis:}y=-\sqrt{x}y=-\sqrt[3]{x}
\text{Reflection across the }y\text{-axis:}y=\sqrt{-x}y=\sqrt[3]{-x}
\text{Vertical stretch } \left|a\right|>1 \text{ or vertical shrink } 0<\left|a\right|<1 \text{:}y=a\sqrt{x}y=a\sqrt[3]{x}
\text{Horizontal translation by } h \\ \text{Vertical translation by } k \text{:}y=\sqrt{x-h} + ky=\sqrt[3]{x-h} + k
(h, k) \text{:}\text{Endpoint}\text{Point of inflection}

The domain and range of the square root function will change with a reflection, or as h or k changes, while the domain and range of the cube root function will continue to be all real numbers.

Examples

Example 1

Consider the following function:f\left(x\right) = -\sqrt{x + 2}

a

Describe the transformation that occurred to y=\sqrt{x} to give f\left(x\right).

Worked Solution
Create a strategy

The function is of the form f\left(x\right)=a\sqrt{x-h} with a=-1 and h=-2.

Apply the idea

The function has been translated to the left by 2 units and reflected across the x-axis.

b

Draw a graph of the function.

Worked Solution
Create a strategy

We will start by completing a table of values for the parent function, y=\sqrt{x}. To do this, we can use the fact that the inverse is \begin{aligned}y=\sqrt{x}\\x=\sqrt{y}\\x^2=y\end{aligned} Since the range of y=\sqrt{x} is x\geq 0, this will be the domain of the inverse, y=x^2. Building a table for the inverse, we have

x01234
y=x^2014916

Swapping these x- and y-values gives us the key points of the parent function, y=\sqrt{x}.

Apply the idea

Using the key points of the parent function, y=\sqrt{x}, with the transformations identified in part (a), we can identify the key points of the given function.

The key points of the parent function are shown in the table below.

x014916
y=\sqrt{x}01234

Reflecting across the x-axis results in:

x014916
y=-\sqrt{x}0-1-2-3-4

Translating these points left 2 units gives us:

x-2-12714
f\left(x\right)=-\sqrt{x+2}0-1-2-3-4

Now, we can graph the function using these key points.

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c

State the domain and range of f\left(x\right).

Worked Solution
Create a strategy

As the graph has been translated 2 units to the left, the domain will change. The range will also change since the function was reflected across the x-axis.

Apply the idea

Domain: \left[-2, \infty\right)

Range: \left(- \infty,0\right]

d

Determine the inverse of f\left(x\right).

Worked Solution
Create a strategy

To find the inverse, we need to write f\left(x\right) as y, swap x and y, then solve for y. After, we need to determine whether the domain of the inverse matches the range of the original function and whether the range of the inverse matches the domain of the original function.

Apply the idea

First, let's find an expression for f^{-1} \left( x \right).

\displaystyle y\displaystyle =\displaystyle -\sqrt{x+2}Write f\left(x\right) and y

Inverse:

\displaystyle x\displaystyle =\displaystyle -\sqrt{y+2}Swap x and y
\displaystyle -x\displaystyle =\displaystyle \sqrt{y+2}Multiply both sides by -1
\displaystyle x^2\displaystyle =\displaystyle y+2Square both sides
\displaystyle x^2-2\displaystyle =\displaystyle ySubtract 2 from both sides

From here, we can see that the result is a quadratic which would form a parabola when graphed. The domain of a quadratic is all real numbers which does not match the range of the original function.

Because the range of the original function is \left(-\infty,0\right], we need to restrict the domain of the inverse to match. Therefore, f^{-1}\left(x\right)=x^2-2 for x\leq 0.

e

Graph the inverse.

Worked Solution
Create a strategy

The inverse, y=x^2 -2, will have a domain of \left(-\infty,0\right] and a range of [-2, \infty).

Apply the idea
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Example 2

Consider the following function:f \left( x \right) = - \sqrt[3]{x}+3

a

Describe the transformation that occurred to y=\sqrt[3]{x} to give f\left(x\right).

Worked Solution
Create a strategy

The function is of the form f\left(x\right)=a\sqrt[3]{x}+k with a=-1, k=3.

Apply the idea

The function has been reflected about the x-axis and translated up by 3 units.

b

Draw a graph of the function.

Worked Solution
Create a strategy

We will start by completing a table of values for the parent function, y=\sqrt[3]{x}. To do this, we can use the fact that the inverse is \begin{aligned}y=\sqrt[3]{x}\\x=\sqrt[3]{y}\\x^3=y\end{aligned} Since y=\sqrt[3]{x} is one-to-one, we do not have to restrict the domain of the inverse. Building a table for the inverse, we have

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y=x^3-8-1018

Swapping these x- and y-values gives us the key points of the parent function, y=\sqrt[3]{x}.

Apply the idea

Using the key points of the parent function, y=\sqrt[3]{x}, with the transformations identified in part (a), we can identify the key points of the given function.

The key points of the parent function are shown in the table below.

x-8-1018
y=\sqrt[3]{x}-2-1012

Reflecting across the x-axis results in:

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y=-\sqrt[3]{x}210-1-2

Translating these points up 3 units gives us:

x-8-1018
y=-\sqrt[3]{x}+354321

Now, we can graph the function using these key points.

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c

State the domain and range of f\left(x\right).

Worked Solution
Create a strategy

The graph has been reflected across the x-axis and translated 3 units up. As the function is a cube root function, the domain and range of the parent function was all real x and y. These transformations do not change them.

Apply the idea

Domain: \left(-\infty, \infty\right)

Range: \left(-\infty, \infty\right)

Example 3

Graph the following piecewise function:

f\left(x\right) = \begin{cases} \sqrt[3]{x-3}, & \enspace x \geq 0 \\ \dfrac{x}{4} + 2 , & \enspace x < 0 \end{cases}

Worked Solution
Create a strategy

We can start by graphing each function on the same coordinate plane. Then, we can restrict the domain based on the piecewise function.

The first equation listed is a cube root function. It is in the form y=\sqrt[3]{x-h} which tells us it has been translated. Since h=3, it has shifted 3 units to the right.

The second equation listed is a linear function with a slope of \dfrac{1}{4} and a y-intercept at \left(0,2\right).

Apply the idea

In the previous problem, we found the key points of the parent cube root function, y=\sqrt[3]{x}. The key points are shown in the table below.

x-8-1018
y=\sqrt[3]{x}-2-1012

Shifting these values right 3 units results in:

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y=\sqrt[3]{x-3}-2-1012

Graphing this and the linear function:

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Restricting the domain:

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Idea summary

Refer to transformations of radical functions with the following table:

Square rootCube root
\text{Parent function:}y= \sqrt{x}y= \sqrt[3]{x}
\text{Reflection across the }x\text{-axis:}y=-\sqrt{x}y=-\sqrt[3]{x}
\text{Reflection across the }y\text{-axis:}y=\sqrt{-x}y=\sqrt[3]{-x}
\text{Stretch } \left|a\right|>1 \text{ or shrink } 0<\left|a\right|<1 \text{:}y=a\sqrt{x}y=a\sqrt[3]{x}
\text{Horizontal translation by } h \\ \text{Vertical translation by } k \text{:}y=\sqrt{x-h} + ky=\sqrt[3]{x-h} + k
(h, k) \text{:}\text{Endpoint}\text{Point of inflection}

The domain and range of the square root function will change with a reflection, or as h or k changes, while the domain and range of the cube root function will continue to be all real numbers.

Outcomes

F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.C.7.B

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

F.BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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