Chapter 5: Q 56. (page 257)

In Problems 53–58, find functionsf and g so that $f \circ g = H$ .
role="math" localid="1646319984809" $H (x) = \sqrt{1 - x^{2}}$

Short Answer

Expert verified

The required functions are $f (x) = \sqrt{x}; g (x) = 1 - x^{2}$ .

Step by step solution

Step 1. Given information.

The given function is:

H (x) = \sqrt{1 - x^{2}}

In the given function H takes $1 - x^{2}$ and raises to the power $\frac{1}{2}$ .

Now decompose H by raising $g (x) = 1 - x^{2}$ to the power $\frac{1}{2}$ .

Let's take $g (x) = 1 - x^{2}$ and $f (x) = \sqrt{x}$ .

Step 2. Find f∘g.

$(f \circ g) (x) = f (g (x))$

Substitute $g (x) = 1 - x^{2}$ in the function $f (g (x))$ ,

Then the function will become $f (1 - x^{2})$ .

Now replace x with $1 - x^{2}$ in $f (x) = \sqrt{x}$ ,

$f (1 - x^{2}) = \sqrt{1 - x^{2}} \sqrt{1 - x^{2}} = H (x)$

As we can see that $f \circ g = H$ , therefore the values of the function that we assumed are correct.

$f (x) = \sqrt{x}; g (x) = 1 - x^{2}$

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.

In Problems 53–58, find functionsf and g so that $f \circ g = H$ .
role="math" localid="1646319984809" $H (x) = \sqrt{1 - x^{2}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find f∘g.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Applied Mathematics

Discrete Mathematics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 53–58, find functionsf and g so that f∘g=H.role="math" localid="1646319984809" H(x)=1-x2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find f∘g.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Applied Mathematics

Discrete Mathematics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 53–58, find functionsf and g so that $f \circ g = H$ .
role="math" localid="1646319984809" $H (x) = \sqrt{1 - x^{2}}$