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Chapter 1: Problem 42

Find (a) \(f \circ g\) and (b) \(g \circ f .\) Find the domain of each function and each composite function. $$f(x)=\frac{3}{x^{2}-1}, \quad g(x)=x+1$$

Short Answer

Expert verified
\(f \circ g(x) = \frac{3}{x^2 + 2x}\) with domain \(\(-\infty, -2) \cup (-2, 0) \cup (0, \infty)\), \(g \circ f(x) = \frac{3}{x^2 - 1} + 1\) with domain \(\(-\infty, -1) \cup (-1, 1) \cup (1, \infty)\)

Step by step solution

01

Compute \(f \circ g(x)\)

To compute this, we replace every instance of \(x\) in \(f\) with \(g(x)\). This would give: \(f \circ g(x) = \frac{3}{(g(x))^2 - 1} = \frac{3}{(x+1)^2 - 1}\). Simplifying this we get, \(f \circ g(x) = \frac{3}{x^2 + 2x + 1 - 1} = \frac{3}{x^2 + 2x}\).
02

Compute \(g \circ f(x)\)

To compute this, we replace every instance of \(x\) in \(g\) with \(f(x)\). This would give: \(g \circ f(x) = f(x) + 1 = \frac{3}{x^2 - 1} + 1\).
03

Find the Domain of \(f(x)\)

The denominator cannot be zero so \(x^{2} - 1 \neq 0\). Solving this inequality yields \(x \neq 1\) and \(x \neq -1\). So the domain of \(f(x)\) is \(\(-\infty, -1) \cup (-1, 1) \cup (1, \infty)\)
04

Find the Domain of \(g(x)\)

The function \(g(x) = x+1\) is defined for all real values of \(x\). Therefore, the domain is \(\(-\infty, \infty)\)
05

Find the Domain of \(f \circ g(x)\)

The denominator in the function \(f \circ g(x) = \frac{3}{x^2 + 2x}\) cannot be zero. Solving \(x^2 + 2x = 0\) we find that \(x=0\) and \(x=-2\). The domain is \(\(-\infty, -2) \cup (-2, 0) \cup (0, \infty)\)
06

Find the Domain of \(g \circ f(x)\)

The function \(g \circ f(x)\) is defined for all real values of \(x\) except for the restrictions imposed by \(f(x)\). Therefore, the domain of \(g \circ f(x)\) is the same as the domain of \(f(x)\), which is \(\(-\infty, -1) \cup (-1, 1) \cup (1, \infty)\)

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Most popular questions from this chapter

Determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped.

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