$$\{x \mid x \geq 0, x \neq 4\}$$

$$(j \circ g)(x) = \sqrt{x} - 1$$, \quad \text{Domain: } x \geq 0$

$$(f \circ g)(x) = \frac{1}{x}$$, \quad \text{Domain: } x \neq 0$

Question 1

Let $$f(x) = x^2$ + 2$$ and $$g(x) = rac{1}{x-1}. $$What is the domain of the composition $$(g \circ f)(x)$$?

Accepted Answer

$$\{x \mid x^2$ + 2 
eq 1\}$$

Question 2

Given $$f(x) = \sqrt{x}$$ and $$g(x) = x - 1$$, find and simplify the formula for $$(f \circ g)(x).$$

Accepted Answer

$$\sqrt{x-1}$$

Question 3

Let $$f(x) = \frac{1}{x-2}$$ and $$g(x) = \sqrt{x}. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$\{x \mid x \geq 0, x 
eq 4\}$$

Question 4

If $$f(x) = x^2$ + 7$$ and $$g(x) = \sqrt{x}$$, decompose $$y = \sqrt{x+7}$$ into $$f$$ and $$g so $$that $$y = g(f(x)). $$What is $$f(x)$$?

Accepted Answer

$$x+7$$

Question 5

Given $$f(x) = \frac{1}{x-1}$$, $$g(x) = \sqrt{x}$$, and $$j(x) = x-1$$, find $$(j \circ g)(x)$$ and its domain.

Accepted Answer

$$(j \circ g)(x) = \sqrt{x} - 1$$, \quad 	ext{Domain: } x \geq 0$

Question 6

Let $$f(x) = \frac{1}{x-1}$$ and $$g(x) = x+1. $$Find the formula for $$(f \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = \frac{1}{x}$$, \quad 	ext{Domain: } x 
eq 0$

Question 7

Suppose $$f(x) = x^2$ + 2$$ and $$g(x) = e^x. $$Decompose $$y = e$$^{x^2 + 2}$$ into f$ and g$ so that y = g(f(x$$))$$. What is g(x$$)$$?

Accepted Answer

$$e^x$$

Question 8

Given $$f(x) = x^2$ + 2$$ and $$g(x) = 6x + 2$$, decompose $$y = (6x+2)^3$ into f$ and g$ so that y = f(g(x$$))$$. What is f(x$$)$$?

Accepted Answer

$$x^3$$

Question 9

Let $$f(x) = \frac{1}{x-2}$$ and $$g(x) = \sqrt{x}. $$Evaluate $$(f \circ g)(25).$$

Accepted Answer

$$\frac{1}{\sqrt{25} - 2}$$

Question 10

Given $$f(x) = \frac{1}{x-1}$$ and $$g(x) = x+1$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$\{x \mid x 
eq 0\}$$

Question 11

Let $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ + 1. $$Find the domain of $$(f \circ g)(x).$$

Accepted Answer

$$\mathbb{R}$$

Question 12

Given $$f(x) = x^2$ + 2$$ and $$g(x) = e^x$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$(f \circ g)(x) = (e^x)^2$ + 2$$

Question 13

Let $$f(x) = x^2$ + 2$$ and $$g(x) = 6x + 2. $$What is $$(g \circ f)(x)$$?

Accepted Answer

$$6(x^2 + 2) + 2$

Question 14

Given $$f(x) = x^2$ + 2$$ and $$g(x) = e^x$$, what is $$(g \circ f)(x)$$?

Accepted Answer

$$e^{x^2 + 2}$$

Question 15

Let $$f(x) = \frac{1}{x-2}$$ and $$g(x) = \sqrt{x}. $$What is $$(f \circ g)(x)$$?

Accepted Answer

$$\frac{1}{\sqrt{x} - 2}$$

Question 16

Given $$f(x) = x^2$ + 2$$ and $$g(x) = 6x + 2$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$(6x + 2)^2$ + 2$$

Question 17

Let $$f(x) = x^2$ + 2$$ and $$g(x) = e^x. $$What is the value of $$(g \circ f)(2)$$?

Accepted Answer

$$e^{2^2 + 2}$$ = $$e^6$$

Question 18

Given $$f(x) = x^2$ + 2$$ and $$g(x) = 6x + 2$$, what is the value of $$(f \circ g)(1)$$?

Accepted Answer

$$(6 \cdot 1 + 2)^2$ + 2 = 8^2$ + 2 = 64 + 2 = 66$$

Function Composition and Decomposition: Domains and Formulas

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