Chapter 5: Q. 37 (page 257)

In Problems 29 – 44, for the given functions f and g, find:
(a) $f o g$
(b) $g o f$
(c) $f o f$
(d) $g o g$
State the domain of each composite function.
$f (x) = \frac{x}{x - 1} g (x) = - \frac{4}{x}$

Short Answer

Expert verified

a) $f \circ g = \frac{4}{4 + x}$ and its domain is: ${x | x \neq 0, x \neq - 4}$

b) $g \circ f = - \frac{4 (x - 1)}{x}$ , and its domain: $\{x | x \neq 1, x \neq 0\}$

c) $f \circ f = x$ and its domain is: $\{x | x \neq 1\}$ .

d) $g \circ g = x$ and its domain is: $\{x | x \neq 0\}$

Step by step solution

Step 1. Given information:

The functions are:

$f (x) = \frac{x}{x - 1} g (x) = - \frac{4}{x}$

The domain of $f (x)$ is: ${x | x \neq 1}$

The domain of $g (x)$ is: ${x | x \neq 0}$

Part (a) Step 1. To find f ∘g and its domain:

$f \circ g (x) = f (g (x)) = \frac{\frac{- 4}{x}}{- \frac{4}{x} - 1} = \frac{- 4}{- 4 - x} = \frac{4}{4 + x}$

The domain of $f \circ g$ is: ${x | x \neq - 4, x \neq 0}$

Part (b) Step 1. To find g∘ f and its domain:

$g \circ f = g (f (x)) = - \frac{4}{\frac{x}{x - 1}} = - \frac{4 (x - 1)}{x}$

The domain of $g \circ f$ is: $\{x | x \neq 0, x \neq 1\}$ .

Part (c) Step 1. To find f∘f and its domain:

$f \circ f = f (f (x)) = \frac{\frac{x}{x - 1}}{\frac{x}{x - 1} - 1} = \frac{\frac{x}{x - 1}}{\frac{x - x + 1}{x - 1}} = \frac{x}{x - 1} \times \frac{x - 1}{1} = x$

Its domain is: $\{x | x \neq 1\}$

Part (c) Step 1. To find g∘g and its domain:

$g \circ g = - \frac{4}{- \frac{4}{x}} = x$

its domain is: $\{x | x \neq 0\}$

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In Problems 29 – 44, for the given functions f and g, find:
(a) $f o g$
(b) $g o f$
(c) $f o f$
(d) $g o g$
State the domain of each composite function.
$f (x) = \frac{x}{x - 1} g (x) = - \frac{4}{x}$

Short Answer

Step by step solution

Step 1. Given information:

Part (a) Step 1. To find f ∘g and its domain:

Part (b) Step 1. To find g∘ f and its domain:

Part (c) Step 1. To find f∘f and its domain:

Part (c) Step 1. To find g∘g and its domain:

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In Problems 29 – 44, for the given functions f and g, find:(a)fog(b) gof(c) fof(d) gogState the domain of each composite function. f(x)=xx-1g(x)=-4x

Short Answer

Step by step solution

Step 1. Given information:

Part (a) Step 1. To find f ∘g and its domain:

Part (b) Step 1. To find g∘ f and its domain:

Part (c) Step 1. To find f∘f and its domain:

Part (c) Step 1. To find g∘g and its domain:

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

In Problems 29 – 44, for the given functions f and g, find:
(a) $f o g$
(b) $g o f$
(c) $f o f$
(d) $g o g$
State the domain of each composite function.
$f (x) = \frac{x}{x - 1} g (x) = - \frac{4}{x}$