Q: A car’s value depreciates exponentially according to $$V(t) = 20000 \times 0.85^t$$, where $$t is $$the number of years after purchase. What is the value after 3 years?

$$V(3) = 20000 \times 0.85^3$ = 12251.25$$

Question 1

Which of the following best describes a function that is one-to-one?

Accepted Answer

Each input is paired with exactly one unique output, and each output is paired with exactly one unique input.

Question 2

Given the function $$f(x) = 2x^2$ - 8x + 6$$, what is the vertex of its graph?

Accepted Answer

$$(2, -2)$$

Question 3

A population of bacteria doubles every 3 hours. If the initial population is 500, which equation models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P = 500 	imes 2$$^{t/3}$$

Question 4

Which transformation will shift the graph of $$f(x) = x^2$ up by 5 units and to the right by 2 units?

Accepted Answer

$$f(x) = (x-2)^2$ + 5$$

Question 5

Given the rational function $$f(x) = \frac{x^2$ - 9}{x^2$ - 4}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 6

A company’s revenue $$R (in $$thousands of dollars) is modeled by $$R(x) = 5x + 20$$, where $$x is $$the number of units sold. What is the slope and what does it represent?

Accepted Answer

Slope is 5; it represents the increase in revenue per unit sold.

Question 7

Which of the following is the domain of the function $$f(x) = \frac{1}{x-3}$$?

Accepted Answer

All real numbers except $$x = 3$$

Question 8

Given the polynomial $$f(x) = x^3$ - 4x^2$ + 4x$$, which of the following is NOT a real zero?

Accepted Answer

$$x = -2$$

Question 9

If $$f(x) is a $$function such that $$f(-x) = f(x)$$ for all $$x in $$its domain, which property does $$f(x)$$ have?

Accepted Answer

It is an even function.

Question 10

A quadratic function has its vertex at $$(3, -5)$$ and passes through the point $$(4, -2). $$What is its equation?

Accepted Answer

$$f(x) = (x-3)^2$ - 5 + 3(x-3)$$

Question 11

Which of the following is the correct equation for the horizontal asymptote of $$f(x) = \frac{2x^2$ + 1}{x^2$ - 3}$$?

Accepted Answer

$$y = 2$$

Question 12

Given the function $$f(x) = 4x - 7$$, evaluate $$f(3).$$

Accepted Answer

$$5$$

Question 13

Which of the following graphs represents a polynomial of degree 4 with three real zeros?

Accepted Answer

A graph crossing the x-axis three times and having four turning points.

Question 14

Which of the following is NOT a property of exponential functions?

Accepted Answer

The function is symmetric about the y-axis.

Question 15

A rational function $$f(x)$$ has a vertical asymptote at $$x = 1$$ and a hole at $$x = 2. $$Which could be its equation?

Accepted Answer

$$f(x) = \frac{(x-2)(x+3)}{(x-1)(x-2)}$$

Question 16

Which of the following is the correct transformation to reflect the graph of $$f(x) = x^3$ over the x-axis?

Accepted Answer

$$f(x) = -x^3$

Question 17

If $$f(x) = \log_2$(x)$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x \u003e 0$$

Question 18

A car’s value depreciates exponentially according to $$V(t) = 20000 	imes 0.85^t$$, where $$t is $$the number of years after purchase. What is the value after 3 years?

Accepted Answer

$$V(3) = 20000 	imes 0.85^3$ = 12251.25$$

Question 19

Which theorem guarantees that a polynomial of degree $$n$$ has exactly $$n$$ complex zeros?

Accepted Answer

The Fundamental Theorem of Algebra

Precalculus Final Exam Study Guide: Functions, Graphs, and Applications

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