Question 1

A radioactive substance has a half-life of 12 hours. If 80 milligrams are present now, how much will be present in 36 hours? (Round your answer to three decimal places.)

Accepted Answer

$$10.000$$

Question 2

The half-life of a substance is 24 hours. If 32 milligrams are present now, how much will be present in 72 hours? (Round your answer to three decimal places.)

Accepted Answer

$$4.000$$

Question 3

A fossilized bone contains 25% of its original amount of carbon-14. How old is the fossil (in years) if the half-life of carbon-14 is 5730 years? Use $$A = A_0$ \left(\frac{1}{2}\$$right)^{t/5730}$$.

Accepted Answer

$$11460$$

Question 4

A thermometer reading $$70^\circ F is $$placed in a cold storage room with a constant temperature of $$40^\circ F. If $$the thermometer reads $$58^\circ F $$after 2 minutes, how long before it will read $$50^\circ F$$? Assume the cooling follows Newton’s Law of Cooling: $$T(t) = T_s + (T_0$ - T_s)e$$^{kt}$$.

Accepted Answer

$$4.8$$ min

Question 5

Which of the following equations best models exponential growth?

Accepted Answer

$$y = 2^x$$

Question 6

Given the function $$f(x) = 2x - 2 + 5$$, what is the domain and range of $$f$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$(-\infty, \infty)$$

Question 7

Which of the following best describes the transformation of $$f(x) = \log_3$(x)$$ compared to $$g(x) = \log_3$(x-2) + 1$$?

Accepted Answer

Shift right 2 units, up 1 unit

Question 8

Given the table below, determine if the function is linear, exponential, or neither. Write the equation of the function.

$$x$$ | $$f(x)
-1$$ | 5
0 | 8
1 | 11
2 | 14
3 | 17

Accepted Answer

Linear, $$f(x) = 3x + 8$$

Question 9

Expand the logarithmic expression: $$\log_2$(8x)$$

Accepted Answer

$$\log_2$ 8 + \log_2$ x$$

Question 10

Write as a single logarithm: $$2\log_3$ x + \log_3$ y$$

Accepted Answer

$$\log_3$ (x^2$y)$$

Question 11

The logistic growth model $$P(t) = \frac{1200}{1 + 3e$$^{-0.5t}$$}$$ represents the population of a bacteria in a culture tube after $$t$$ hours. What is the approximate population after 4 hours?

Accepted Answer

$$1050$$

Question 12

A thermometer reading $$90^\circ F is $$placed in a room with a constant temperature of $$70^\circ F. If $$the thermometer reads $$80^\circ F $$after 5 minutes, what will it read after 10 minutes? (Use Newton’s Law of Cooling.)

Accepted Answer

$$75^\circ F$$

Question 13

Determine if the following functions are inverses: $$f(x) = 2x + 3$$ and $$g(x) = \frac{x-3}{2}. $$Explain your answer.

Accepted Answer

Yes, because $$f(g(x)) = x$$ and $$g(f(x)) = x$$

Question 14

If a population of bacteria doubles every 3 hours, how long will it take for the population to increase by a factor of 8?

Accepted Answer

$$9$$ hours

Question 15

Which of the following is the correct domain for the function $$f(x) = \log_3$(x)$$?

Accepted Answer

$$(0, \infty)$$

Question 16

A certain medicine decays in the bloodstream according to $$A(t) = A_0$ e$$^{-0.2t}$$. If the initial amount is 100 mg, how much remains after 5 hours?

Accepted Answer

$$36.79 mg$$

Question 17

Which of the following best describes the range of $$f(x) = 2x - 2 + 5$$?

Accepted Answer

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

Question 18

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

Accepted Answer

$$3(x-5)^2$$

Question 19

Which of the following is the correct equation for Newton’s Law of Cooling?

Accepted Answer

$$u(t) = T + (u_0$ - T)e$$^{kt}$$

Precalculus Study Guide: Exponential Growth, Decay, and Logistic Models

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