Chapter 5: Q. 54 (page 269)

The function $f$ is one-to-one. Find its inverse and check your answer. Graph $f$ , $f^{- 1}$ , and y = x on the same coordinate axes.
$f (x) = x^{3} + 1$

Short Answer

Expert verified

The inverse of the function is $f^{- 1} (x) = \sqrt[3]{x - 1}$ .

Step by step solution

Step 1. Given information :

The given function is one-to-one function.

$f (x) = x^{3} + 1$ .

Step 2. Finding the inverse of the function :

For finding the inverse of the function we need to interchange the variables.

let, $f (x) = y = x^{3} + 1$

Interchanging the variables x, y,

$x = y^{3} + 1 y^{3} = x - 1 y = \sqrt[3]{x - 1}$

So, the inverse $f^{- 1} (x) = \sqrt[3]{x - 1}$ .

Step 3. Check the solution :

To check whether the function is inverses are not,

let, $f (f^{- 1} (x)) = x$

$f (\sqrt[3]{x - 1}) = x w e g o t, f^{- 1} (x) = \sqrt[3]{x - 1} {(\sqrt[3]{x - 1})}^{3} + 1 = x g i v e n, f (x) = x^{3} + 1 (x - 1) + 1 = x x = x$

Similarly, $f^{- 1} (f (x)) = x$

localid="1647509029875" $f^{- 1} (x^{3} + 1) = x g i v e n, f (x) = x^{3} + 1 \sqrt[3]{(x^{3} + 1) - 1} = x, f^{- 1} (x) = \sqrt[3]{x - 1} \sqrt[3]{x^{3}} = x x = x, f^{- 1} (x) = (- \infty, \infty) h e n c e p r o v e d .$

Step 4. Graph representation of f,f-1 and y=x :

Graph ;

Domain of $f$ = range of localid="1647508985116" $f^{- 1} = (- \infty, \infty)$

Domain of $f^{- 1}$ = range of localid="1647508995977" $f = (- \infty, \infty)$ .

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The function $f$ is one-to-one. Find its inverse and check your answer. Graph $f$ , $f^{- 1}$ , and y = x on the same coordinate axes.
$f (x) = x^{3} + 1$

Short Answer

Step by step solution

Step 1. Given information :

Step 2. Finding the inverse of the function :

Step 3. Check the solution :

Step 4. Graph representation of f,f-1 and y=x :

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The function fis one-to-one. Find its inverse and check your answer. Graph f, f-1, and y = x on the same coordinate axes.f(x)=x3+1

Short Answer

Step by step solution

Step 1. Given information :

Step 2. Finding the inverse of the function :

Step 3. Check the solution :

Step 4. Graph representation of f,f-1 and y=x :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

The function $f$ is one-to-one. Find its inverse and check your answer. Graph $f$ , $f^{- 1}$ , and y = x on the same coordinate axes.
$f (x) = x^{3} + 1$