/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 Let \(g(x)=x^{2}+3 .\) Find a fu... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 50

Let \(g(x)=x^{2}+3 .\) Find a function \(f\) that produces the given composition. $$(f \circ g)(x)=\frac{1}{x^{2}+3}$$

Short Answer

Expert verified
Answer: The function \(f(x)\) is \(f(x)=\frac{1}{x}\).

Step by step solution

01

Write down given information

We are given: - \(g(x)=x^2+3\) - \((f \circ g)(x)=\frac{1}{x^{2}+3}\)
02

Uncover f#g(x)

Rewrite the given composition as: $$f(g(x))=\frac{1}{x^{2}+3}$$ Now, let's express \(g(x)\) in terms of \(y\): $$y = x^2 + 3$$ Now we have: $$f(y)=\frac{1}{y}$$ Since we found the function \(f(y)\), we can rewrite it in terms of \(x\): $$f(x)=\frac{1}{x}$$
03

Confirm the Composition#g(x)

To check if our found function \(f(x)\) indeed gives the required composition, let's calculate \((f \circ g)(x)\): $$\begin{aligned}(f \circ g)(x) &= f(g(x))\\ &= f(x^2+3)\\ &= \frac{1}{x^2+3}\end{aligned}$$ As we can see, we indeed found the correct function \(f(x)\) to produce the given composition \((f \circ g)(x)=\frac{1}{x^{2}+3}\).

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