Question 1

Given $$f(x) = 2x^2$ - 3$$ and $$g(x) = 4x$$, what is $$(f \circ g)(1)$$?

Accepted Answer

$$f(g(1)) = 2(4)^2$ - 3 = 32 - 3 = 29$$

Question 2

If $$f(x) = \dfrac{8}{x+2}$$ and $$g(x) = \sqrt[3]{x}$$, what is $$(f \circ g)(8)$$?

Accepted Answer

$$f(g(8)) = \dfrac{8}{\sqrt[3]{8} + 2} = \dfrac{8}{2 + 2} = 2$$

Question 3

Let $$f(x) = x^2$ + 3x - 1$$ and $$g(x) = 2x + 3. $$What is the formula for $$(f \circ g)(x)$$?

Accepted Answer

$$f(g(x)) = (2x+3)^2$ + 3(2x+3) - 1 = 4x^2$ + 12x + 9 + 6x + 9 - 1 = 4x^2$ + 18x + 17$$

Question 4

Given $$f(x) = \dfrac{1}{x+2}$$ and $$g(x) = \dfrac{4}{x-1}$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x 
eq 1,\ x 
eq -1$$

Question 5

Suppose $$f(x) = \sqrt{x}$$ and $$g(x) = 2x + 5. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \geq -2.5$$

Question 6

If $$f(x) = \dfrac{1}{x+2}$$, what is $$(f \circ f)(x)$$ and its domain?

Accepted Answer

$$(f \circ f)(x) = \dfrac{1}{\dfrac{1}{x+2} + 2} = \dfrac{x+2}{2x+5}$$, domain: $$x 
eq -2,\ x 
eq -\dfrac{5}{2}$$

Question 7

Given $$f(x) = x^2$ + 3x - 1$$ and $$g(x) = 2x + 3$$, what is the domain of $$(g \circ f)(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 8

If $$f(x) = \dfrac{4}{x-1} + 2$$ and $$g(x) = 2x + 5$$, what is $$(f \circ g)(x)$$ and its domain?

Accepted Answer

$$(f \circ g)(x) = \dfrac{4}{2x+5-1} + 2 = \dfrac{4}{2x+4} + 2$$, domain: $$x 
eq -2$$

Question 9

Which of the following best describes the process of finding the domain of $$(f \circ g)(x)$$?

Accepted Answer

Exclude any $$x$$ not in the domain of $$g$$, and any $$x$$ for which $$g(x) is $$not in the domain of $$f.$$

Question 10

Let $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 4. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \leq -2 or$$ $$x \geq 2$$

Question 11

Suppose $$f(x) = \dfrac{1}{x}$$ and $$g(x) = x^2$ - 9. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x 
eq 3,\ x 
eq -3$$

Question 12

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is (f$$ \circ g)(x)$$ and its domain?

Accepted Answer

$$(f \circ g)(x) = (\sqrt{x})^2$ = x$$, domain: $$x \geq 0$$

Question 13

Given $$f(x) = \dfrac{1}{x+2}$$ and $$g(x) = \dfrac{4}{x-1}$$, what is $$(g \circ f)(x)$$ and its domain?

Accepted Answer

$$(g \circ f)(x) = \dfrac{4}{\dfrac{1}{x+2} - 1}$$, domain: $$x 
eq -2,\ x 
eq -1$$

Question 14

A medical test uses the function $$f(x) = \dfrac{1}{x-1} to $$model the risk of a disease based on a patient's blood marker $$x. If $$the marker is itself a function of age, $$g(a) = 2a + 3$$, what is the risk function in terms of age, and for which ages is it defined?

Accepted Answer

$$f(g(a)) = \dfrac{1}{2a+3-1} = \dfrac{1}{2a+2}$$, defined for $$a 
eq -1$$

Question 15

If $$f(x) = \sqrt{x+2}$$ and $$g(x) = x^2$ - 1$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \leq -\sqrt{3} or$$ $$x \geq \sqrt{3}$$

Question 16

Given $$f(x) = \dfrac{1}{x}$$ and $$g(x) = \dfrac{1}{x}$$, what is $$(f \circ g)(x)$$ and its domain?

Accepted Answer

$$(f \circ g)(x) = f\left(\dfrac{1}{x}ight) = x$$, domain: $$x 
eq 0,\ x 
eq 1$$

Question 17

A chemical reaction rate is modeled by $$f(x) = \sqrt{x}$$, where $$x is $$the concentration of a reactant. If the concentration changes over time as $$g(t) = t^2$ + 1$$, what is the reaction rate as a function of time, and for which $$t is it $$defined?

Accepted Answer

$$f(g(t)) = \sqrt{t^2$ + 1}$$, defined for all real $$t$$

Composite Functions: Formation, Evaluation, and Domain

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