Question 1

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

Accepted Answer

$$(2x - 3)^2$ + 1$$

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 $$f(x) = \frac{1}{x-2}$$ 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

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

Accepted Answer

$$x = -3$$

Question 5

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

Accepted Answer

$$(g \circ f)(x) = |x|$$, domain: $$(-\infty, \infty)$$

Question 6

Suppose $$f(x) = \frac{1}{x+4}$$ and $$g(x) = \frac{1}{x-1}. $$What value(s) must be excluded from the domain of $$(f \circ g)(x)$$?

Accepted Answer

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

Question 7

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

Accepted Answer

$$(2x + 1)^3$

Question 8

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

Accepted Answer

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

Question 9

A real-world scenario: The temperature $$T (in$$ °C) at time $$t (in $$hours) is given by $$T(t) = 5t + 10. $$The energy consumption $$E (in $$kWh) as a function of temperature is $$E(T) = 2T^2$. What is the composite function E(T(t$$))$$?

Accepted Answer

$$2(5t + 10)^2$

Question 10

Suppose $$f(x) = x^2$ + 2$$ and $$g(x) = \sqrt{x}. $$Find $$(f \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = x + 2$$, domain: $$[0, \infty)$$

Question 11

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

Accepted Answer

$$(f \circ g)(x) = x$$, domain: $$(-\infty, 0) \cup (0, \infty)$$, x 
eq 1$

Question 12

Given $$f(x) = x^2$ and g(x) = x + 1$, show that (f$$ \circ g)(x) = (g \circ f)(x)$$ for all x$ in the domain.

Accepted Answer

False, because $$(f \circ g)(x) = (x + 1)^2$ and (g$$ \circ f)(x) = $$x^2 + 1$

Question 13

Find functions $$f$$ and $$g$$ such that $$H(x) = (3x + 2$$)^{21}$$ and H(x) = f(g(x$$))$$.

Accepted Answer

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

Question 14

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

Accepted Answer

$$(-\infty, -3] \cup [3, \infty)$$

Question 15

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

Accepted Answer

$$x 
eq -2,\ x 
eq -1$$

Question 16

A medication's effectiveness $$E (in $$percent) depends on the concentration $$C (in mg/L) as$$ $$E(C) = 100 - 5C^2$. If the concentration C$ at time t$ (in hours) is C(t) = 2t + 1$, what is the composite function E(C(t$$))$$?

Accepted Answer

$$100 - 5(2t + 1)^2$

Composite Functions: Formation, Evaluation, and Domain Analysis

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