Chapter 5: Q.13 (page 349)

The function $f$ is one-to-one. Find the inverse of each function and check your answer. State the
domain and the range of $f$ and $f^{- 1}$ .
$f (x) = x^{\frac{1}{3}} + 1$

Short Answer

Expert verified

The inverse is

$f^{- 1} (x) = {(x - 1)}^{3}$

Step by step solution

Concept used

At first, in $y = f (x)$ , interchange the variables $x$ and $y$ , to obtain $x = f (y)$ .

Then solve the equation for $y$ in terms of $x$ .

After that, check the result by showing that,

$f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$

Interchange the variables x and y

Given that, $f (x) = x^{\frac{1}{3}} + 1$

Replacing $f (x)$ with $y$ and interchange the variables $x$ and $y$ in

$y = x^{\frac{1}{3}} + 1$

to obtain

$x = y^{\frac{1}{3}} + 1$

Solve for y in terms of x

$x = y^{\frac{1}{3}} + 1$

Subtract $1$ from both sides,

$x - 1 = y^{\frac{1}{3}}$

Taking power of $3$ to the both sides,

${(x - 1)}^{3} = y$

$y = {(x - 1)}^{3}$

The inverse is

$f^{- 1} (x) = {(x - 1)}^{3}$

Check result

$f^{- 1} (f (x)) = f^{- 1} (x^{\frac{1}{3}} + 1)$

$= {(x^{\frac{1}{3}} + 1 - 1)}^{3}$

$= {(x^{\frac{1}{3}})}^{3}$

$= x$

$f (f^{- 1} (x)) = f {(x - 1)}^{3}$

$= {\{{(x - 1)}^{3}\}}^{\frac{1}{3}} + 1$

$= (x - 1) + 1$

$= x$

Thus, the result is checked.

Statement of domain and range

$f^{- 1} (f (x)) = x$ , where $x$ is in domain of $f$ and $D o m a i n o f f = R a n g e o f f^{- 1}$

$f (f^{- 1} (x)) = x$ , where $x$ is in the domain of $f^{- 1}$ androle="math" localid="1646449654799" $D o m a i n o f f^{- 1} = R a n g e o f f$

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The function $f$ is one-to-one. Find the inverse of each function and check your answer. State the
domain and the range of $f$ and $f^{- 1}$ .
$f (x) = x^{\frac{1}{3}} + 1$

Short Answer

Step by step solution

Concept used

Interchange the variables x and y

Solve for y in terms of x

Check result

Statement of domain and range

One App. One Place for Learning.

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The function fis one-to-one. Find the inverse of each function and check your answer. State thedomain and the range of fand f-1.fx=x13+1

Short Answer

Step by step solution

Concept used

Interchange the variables x and y

Solve for y in terms of x

Check result

Statement of domain and range

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

The function $f$ is one-to-one. Find the inverse of each function and check your answer. State the
domain and the range of $f$ and $f^{- 1}$ .
$f (x) = x^{\frac{1}{3}} + 1$