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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
Question: Find the function f(x) such that when composed with the function g(x) = x^2 + 3, the result is (f 鈭� g)(x) = 1/(x^2 + 3). Answer: f(x) = 1/x

Step by step solution

01

Evaluate the given function g(x)

The function \(g(x) = x^2 + 3\) has already been provided in the question. So we can use it as is.
02

Express the given composition in terms of g(x)

We want to find a function \(f(x)\) such that \((f \circ g)(x) = \frac{1}{x^2 + 3}\). We can rewrite this equation in terms of \(g(x)\) as follows: $$(f \circ g)(x) = f(g(x)) = \frac{1}{g(x)}$$
03

Determine the function f(x)

Now we need to find a function \(f(x)\) that will produce the desired composition with \(g(x)\). From the equation we derived in Step 2, we see that the composition is equal to \(\frac{1}{g(x)}\). This suggests that we can choose \(f(x)\) to be a function that finds the reciprocal of its input. Therefore, we can define \(f(x)\) as follows: $$f(x) = \frac{1}{x}$$
04

Verify the composition

Let's verify that \((f \circ g)(x) = \frac{1}{x^2 + 3}\) with our chosen functions \(f(x)\) and \(g(x)\). We have: $$(f \circ g)(x) = f(g(x)) = f(x^2 + 3) = \frac{1}{x^2 + 3}$$ This confirms that we have found the correct functions to produce the given composition. In conclusion, the function \(f(x)\) needed to produce the given composition is $$f(x) = \frac{1}{x}$$.

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