Q: A medical researcher models the concentration $$C of a $$drug in the bloodstream as $$C(t) = 5e$$^{-0.2t}$$, where t$ is time in hours. What is the concentration after 3$ hours?

$$C(3) = 5e$$^{-0.6}$$ \\approx 2.74$$

Q: Given $$f(x) = 4x - 1$$, find the inverse function $$f^{-1}(x$$)$$.

$$f^{-1}(x$$) = \\frac{x + 1}{4}$$

Q: A population of bacteria doubles every $$4$$ hours. If the initial population is $$500$$, what is the population after $$12$$ hours?

$$P = 500 \\times 2$$^{3}$$ = 4000$$

Q: A function $$f(x) is $$even if:

$$f(-x) = f(x)$$ \\text{ for all } x$

Question 1

A company models its profit $$P (in $$thousands of dollars) as a function of time $$t (in $$months) by $$P(t) = -2t^2$ + 24t - 40. $$What is the maximum profit, and after how many months does it occur?

Accepted Answer

Maximum profit is $$104$$ thousand dollars at $$t = 6$$ months.

Question 2

Given the function $$f(x) = 3x - 7$$, what is the $$y-$$intercept of its graph?

Accepted Answer

$$(0, -7)$$

Question 3

Which of the following describes the transformation applied to $$f(x) to $$obtain $$g(x) = f(x + 4) - 2$$?

Accepted Answer

Shift left by $$4$$ units and down by $$2$$ units.

Question 4

A function $$h(x) is $$defined as $$h(x) = |x - 3| + 2. $$What is the vertex of the graph of $$h(x)$$?

Accepted Answer

$$(3, 2)$$

Question 5

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

Accepted Answer

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

Question 6

A medical researcher models the concentration $$C of a $$drug in the bloodstream as $$C(t) = 5e$$^{-0.2t}$$, where t$ is time in hours. What is the concentration after 3$ hours?

Accepted Answer

$$C(3) = 5e$$^{-0.6}$$ \approx 2.74$$

Question 7

Which of the following functions is one-to-one?

Accepted Answer

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

Question 8

Given $$f(x) = 4x - 1$$, find the inverse function $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x$$) = \frac{x + 1}{4}$$

Question 9

A line passes through the points $$(2, 5)$$ and $$(6, 13). $$What is the equation of the line in point-slope form?

Accepted Answer

$$y - 5 = 2(x - 2)$$

Question 10

Which of the following describes the graph of $$y = -f(x)$$ compared to $$y = f(x)$$?

Accepted Answer

It is reflected over the $$x-$$axis.

Question 11

A car rental company charges a flat fee of $$\30$ plus $$\$$0.20$$ per mile driven. Write a function $$C(m)$$ representing the total cost for $$m$$ miles, and find the cost for $$150$$ miles.

Accepted Answer

$$C(m) = 30 + 0.2m$$, so $$C(150) = 60$$

Question 12

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

Accepted Answer

All real numbers except $$x = 4$$

Question 13

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

Accepted Answer

$$(f + g)(x) = x^2$ + 2x$$

Question 14

A function $$f(x) is $$decreasing on the interval $$(-\infty, 2)$$ and increasing on $$(2, \infty). $$What can be said about $$x = 2$$?

Accepted Answer

$$x = 2 is a $$local minimum.

Question 15

Which of the following is the equation of a line parallel to $$y = -3x + 7$$ and passing through $$(2, -1)$$?

Accepted Answer

$$y + 1 = -3(x - 2)$$

Question 16

A population of bacteria doubles every $$4$$ hours. If the initial population is $$500$$, what is the population after $$12$$ hours?

Accepted Answer

$$P = 500 \times 2$$^{3}$$ = 4000$$

Question 17

Which of the following is the correct order to apply transformations to the function $$f(x)$$: vertical stretch, horizontal shift, reflection over $$x-$$axis?

Accepted Answer

Horizontal shift, vertical stretch, reflection over $$x-$$axis.

Question 18

Given $$f(x) = x^2$ - 4x + 3$$, what are the $$x-$$intercepts of the graph?

Accepted Answer

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

Question 19

A function $$f(x) is $$even if:

Accepted Answer

$$f(-x) = f(x)$$ \text{ for all } x$

Precalculus Study Guide: Functions, Graphs, and Transformations

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