Question 1

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

Accepted Answer

$$(f \circ g)(x) = (2x - 3)^2$ + 1$$

Question 2

If $$f(x) = \sqrt{x}$$ and $$g(x) = x - 4$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$[4, \infty)$$

Question 3

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

Accepted Answer

$$(-\infty, -\sqrt{2}) \cup (\sqrt{2}, \infty)$$

Question 4

Given $$f(x) = x + 5$$ and $$g(x) = \sqrt{x}$$, find $$(g \circ f)(x)$$ and its domain.

Accepted Answer

$$(g \circ f)(x) = \sqrt{x + 5}$$, domain: $$[-5, \infty)$$

Question 5

If $$f(x) = \dfrac{1}{x}$$ and $$g(x) = x - 3$$, what value(s) must be excluded from the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x = 3$$

Question 6

Let $$f(x) = x^2$ and g(x$$) = \dfrac{1}{x+1}$$. What is (f$$ \circ g)(x)$$ and its domain?

Accepted Answer

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

Question 7

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

Accepted Answer

$$(-\infty, -\sqrt{2}] \cup [\sqrt{2}, \infty)$$

Question 8

Given $$f(x) = x^3$ and g(x) = 2x + 1$, find (g$$ \circ f)(2)$$.

Accepted Answer

$$(g \circ f)(2) = 17$$

Question 9

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

Accepted Answer

$$(f \circ g)(x) = x$$

Question 10

A chemist models the temperature $$T (in$$ °C) of a reaction as a function of time $$t (in $$minutes) by $$T(t) = 5t + 20. $$The concentration $$C (in $$mol/L) of a reactant is $$C(T) = 100 - 2T. $$What is the composite function $$C(T(t))$$ and what does it represent?

Accepted Answer

$$C(T(t)) = 100 - 2(5t + 20) = 60 - 10t$$, representing concentration as a function of time

Question 11

Let $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 4. $$For which values of $$x is$$ $$(f \circ g)(x)$$ defined?

Accepted Answer

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

Question 12

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

Accepted Answer

$$x 
eq 1$$

Question 13

Given $$f(x) = \dfrac{1}{x+4}$$ and $$g(x) = \dfrac{1}{x-1}$$, what value(s) must be excluded from the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x = 1$$ and $$x = 5$$

Question 14

If $$H(x) = (3x + 2$$)^{21}$$, which of the following is a valid decomposition into f(g(x$$))$$?

Accepted Answer

$$f(x) = x$$^{21}$$,\ g(x) = 3x + 2$$

Question 15

Let $$H(x) = \dfrac{1}{x^2$ + 1}. $$Which pair of functions $$f$$ and $$g$$ satisfies $$H(x) = f(g(x))$$?

Accepted Answer

$$f(x) = \dfrac{1}{x},\ g(x) = x^2$ + 1$$

Question 16

A medical test result $$R$$ depends on the patient's blood sugar $$S by$$ $$R(S) = 0.5S + 10. $$Blood sugar $$S$$ depends on time $$t$$ after a meal by $$S(t) = 80e$$^{-0.1t}$$. What is the composite function R(S(t$$))$$?

Accepted Answer

$$R(S(t)) = 0.5 \cdot 80e$$^{-0.1t}$$ + 10 = 40e$$^{-0.1t}$$ + 10$$

Composite Functions: Definitions, Domains, and Applications

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