Q: A car's value depreciates according to $$V(t) = 20000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value after 5 years?

$$V(5) = 20000 \times (0.85)^5$ \approx 8858$$

Question 1

Which of the following is the correct definition of a function?

Accepted Answer

A relation in which each input is paired with exactly one output.

Question 2

Given the piecewise function $$f(x) = \begin{cases} x^2$ & 	ext{if } x \u003c 0 \ 2x + 1 & 	ext{if } x \geq 0 \end{cases}$$, what is $$f(-3)$$?

Accepted Answer

$$9$$

Question 3

A chemical reaction's rate $$R is $$modeled by the function $$R(t) = 5e$$^{-0.2t}$$, where t$ is time in minutes. What is the rate after 10 minutes?

Accepted Answer

$$R(10) = 5e$$^{-2}$$ \approx 0.676$$

Question 4

Which of the following best describes the domain of the function $$f(x) = \frac{1}{x-4}$$?

Accepted Answer

$$x 
eq 4$$

Question 5

A population of bacteria doubles every 3 hours. If the initial population is 200, which function models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 200 \cdot 2$$^{t/3}$$

Question 6

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

Accepted Answer

$$f(g(2)) = 3 \cdot 4 - 2 = 10$$

Question 7

A medication's concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.5t}$$, where t$ is in hours. After how many hours will the concentration drop to 10$ mg/L?

Accepted Answer

$$t = \frac{\ln(0.2)}{-0.5} \approx 3.22$$

Question 8

Which of the following is the inverse of the function $$f(x) = 2x + 7$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x-7}{2}$$

Question 9

A ball is thrown upward from a height of 2 meters with an initial velocity of 10 m/s. Its height after $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 10t + 2. $$What is the maximum height reached by the ball?

Accepted Answer

$$h_{max} = 7.10$$ meters

Question 10

Which of the following describes the range of the function $$f(x) = \sqrt{x-3}$$?

Accepted Answer

$$[0, \infty)$$

Question 11

If $$f(x) = x^2$ - 4x + 3$$, for which value(s) of $$x$$ does $$f(x) = 0$$?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 12

A car's value depreciates according to $$V(t) = 20000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value after 5 years?

Accepted Answer

$$V(5) = 20000 	imes (0.85)^5$ \approx 8858$$

Question 13

Which of the following is the correct graph for the function $$f(x) = |x-2|$$?

Accepted Answer

A V-shaped graph with vertex at $$(2, 0)$$

Question 14

If $$f(x) = \frac{2x+1}{x-1}$$, what is the vertical asymptote of $$f(x)$$?

Accepted Answer

$$x = 1$$

Question 15

A radioactive substance decays according to $$A(t) = A_0$ e$$^{-kt}$$. If the half-life is 5 years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{5}$$

Question 16

Which of the following is NOT a function?

Accepted Answer

A relation where the input $$x=2 is $$paired with both $$y=3$$ and $$y=5$$

Question 17

If $$f(x) = \sqrt{9 - x^2$}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$-3 \leq x \leq 3$$

Question 18

A company’s profit $$P in $$thousands of dollars is modeled by $$P(x) = -2x^2$ + 12x - 7$$, where $$x is $$the number of units sold in thousands. What is the maximum profit?

Accepted Answer

$$P_{max} = 25$$

Question 19

Which of the following transformations shifts the graph of $$f(x) to $$the right by 4 units?

Accepted Answer

$$f(x-4)$$

Question 20

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

Accepted Answer

$$g(f(-2)) = g(-3) = 9$$

Piecewise Functions and Their Evaluation

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