Chapter 5: Q. 56 (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.
$f (x) = x^{2} + 9 x \geq 0$

Short Answer

Expert verified

The inverse of the function is $f^{- 1} (x) = \sqrt{x - 9}$

Step by step solution

Step 1. Given information :

The given function is one-to-one function.

$f (x) = x^{2} + 9 x \geq 0$

Step 2. Finding the inverse of the function :

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

let, $f (x) = y = x^{2} + 9$

Interchanging the variables x, y,

$x = y^{2} + 9 y^{2} = x - 9 y = \sqrt{x - 9}$

So, the inverse $f^{- 1} (x) = \sqrt{x - 9}$

Step 3. Check the solution :

To check whether the function is inverses are not,

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

$f (\sqrt{x - 9}) = x w e g o t, f^{- 1} (x) = \sqrt{x - 9} {(\sqrt{x - 9})}^{2} + 9 = x g i v e n, f (x) = x^{2} + 9 x - 9 + 9 = x x = x$

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

$f^{- 1} (x^{2} + 9) = x g i v e n, f (x) = x^{2} + 9 \sqrt{(x^{2} + 9) - 9} = x, f^{- 1} (x) = \sqrt{x - 9} \sqrt{x^{2}} = x | x | = x, f^{- 1} (x) = [0, \infty) h e n c e p r o v e d .$

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

Graph ;

Domain of $f$ = range of $f^{- 1} = [9, \infty)$

Domain of $f^{- 1}$ = range of $f = [9, \infty)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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.
$f (x) = x^{2} + 9 x \geq 0$

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

Probability and Statistics

Statistics

Applied Mathematics

Calculus

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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)=x2+9x≥0

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

Probability and Statistics

Statistics

Applied Mathematics

Calculus

Pure Maths

Theoretical and Mathematical Physics

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.
$f (x) = x^{2} + 9 x \geq 0$