Chapter 5: Q. 55 (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.
$f (x) = x^{2} + 4 x \geq 0$

Short Answer

Expert verified

The inverse of the function is $f^{- 1} (x) = \sqrt{x - 4}$ .

Step by step solution

Step 1. Given information :

The given function is one-to-one function.

$f (x) = x^{2} + 4 x \geq 0$

Step 2. Finding the inverse of the function :

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

let, $f (x) = y = x^{2} + 4$

Interchanging the variables x, y,

$x = y^{2} + 4 y^{2} = x - 4 y = \sqrt{x - 4}$

So, the inverse $f^{- 1} (x) = \sqrt{x - 4}$

Step 3. Check the solution :

To check whether the function is inverses are not,

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

localid="1647508144689" $f (\sqrt{x - 4}) = x w e g o t, f^{- 1} (x) = \sqrt{x - 4} {(\sqrt{x - 4})}^{2} + 4 = x g i v e n, f (x) = x^{2} + 4 x - 4 + 4 = x x = x$

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

localid="1647509050463" $f^{- 1} (x^{2} + 4) = x g i v e n, f (x) = x^{2} + 4 \sqrt{(x^{2} + 4) - 4} = x, f^{- 1} (x) = \sqrt{x - 4} \sqrt{x^{2}} = x | x | = x, f^{- 1} (x) = [0, \infty) h e n c e p r o v e d .$

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

Graph ;

Domain of $f$ = range of $f^{- 1} = [4, \infty)$

Domain of $f^{- 1}$ = range of $f = [4, \infty)$

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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.
$f (x) = x^{2} + 4 x \geq 0$

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-1and 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)=x2+4x≥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-1and y=x.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Logic and Functions

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.
$f (x) = x^{2} + 4 x \geq 0$