Chapter 5: Q 55. (page 257)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

The given function is:

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

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

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

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

Step 2. Find f∘g.

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

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

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

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

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

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

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find f∘g.

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In Problems 53–58, find functionsf and g so that f∘g=H.H(x)=x2+1

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

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Geometry

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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