Question 1

Let $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ + 1. $$What is the formula for the composite function $$(f \circ g)(x)$$?

Accepted Answer

$$\sqrt{x^2$ + 1}$$

Question 2

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

Accepted Answer

$$[0,2) \cup (2, \infty)$$

Question 3

If $$f(x) = x^2$ and g(x) = x-3$, what is (g$$ \circ f)(7)$$?

Accepted Answer

$$4$$

Question 4

Suppose $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ + 1. $$Which of the following best describes the process to compute $$(f \circ g)(x)$$?

Accepted Answer

First square $$x$$, add 1, then take the square root

Question 5

Given $$f(x) = \frac{1}{x-2}$$ and $$g(x) = \sqrt{x}$$, what is the domain of $$\left(\frac{f}{g}\right)(x)$$?

Accepted Answer

$$(0,2) \cup (2, \infty)$$

Question 6

If $$f(x) = x^2$ and g(x) = x-3$, what is the formula for (f$$ \circ g)(x)$$?

Accepted Answer

$$(x-3)^2$$

Question 7

A balloon's radius $$R is a $$function of the volume $$V of $$air pumped in, and $$V is a $$function of time $$t. $$Which expression gives the radius as a function of time?

Accepted Answer

$$R(V(t))$$

Question 8

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

Accepted Answer

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

Question 9

Suppose $$f(x) = \sqrt{x+9}. $$Find functions $$g$$ and $$h$$ such that $$f = g \circ h.$$

Accepted Answer

$$g(x) = \sqrt{x},\ h(x) = x+9$$

Question 10

A ship sails at 20 mi/h parallel to a straight shoreline and passes a lighthouse at noon. The distance $$s$$ from the lighthouse after $$t$$ hours is given by $$s = \sqrt{25 + (20t)^2$}. $$What does this function represent?

Accepted Answer

The distance from the lighthouse to the ship as a function of time

Question 11

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

Accepted Answer

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

Question 12

Given $$f(x) = \frac{1}{x-2}$$ and $$g(x) = \sqrt{x}$$, what is $$(f \cdot g)(x)$$?

Accepted Answer

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

Question 13

Which of the following is true about the composition of functions $$f$$ and $$g$$?

Accepted Answer

$$(f \circ g)(x)$$ means apply $$g$$ first, then $$f$$

Question 14

If $$f(x) = x^2$ and g(x) = x-3$, what is (f$$ \circ g)(5)$$?

Accepted Answer

$$4$$

Question 15

Suppose $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ + 1. $$Which of the following is the correct decomposition of $$h(x) = \sqrt{x^2$ + 1}$$?

Accepted Answer

$$h(x) = f(g(x))$$

Question 16

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

Accepted Answer

$$\sqrt{5}$$

Question 17

Given $$f(x) = \frac{1}{x-2}$$ and $$g(x) = \sqrt{x}$$, what is $$(f-g)(x)$$?

Accepted Answer

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

Question 18

Which of the following is a real-world example of function composition?

Accepted Answer

Calculating the distance from a lighthouse to a ship as a function of time, where distance depends on the ship's position, which depends on time

Combining Functions: Sums, Products, Quotients, and Composition

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