Question 1

Which of the following graphs does NOT represent a function? (Refer to the images provided.)

Accepted Answer

The graph of a circle centered at the origin with radius 5.

Question 2

Given the relation $$g = \{ (2, 4), (-1, 3), (-2, 0) \}$$, what is the inverse relation $$g^{-1}$$?

Accepted Answer

$$\{ (4, 2), (3, -1), (0, -2) \}$$

Question 3

Which property is used to determine if a graph represents a function?

Accepted Answer

Vertical-Line Test

Question 4

If $$f(x) = x^2$ - 5x$$, which of the following statements is true about its inverse?

Accepted Answer

The inverse is not a function because some $$x$$ values correspond to multiple $$y$$ values.

Question 5

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

Accepted Answer

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

Question 6

Given $$f(x) = 2x + 3$$, what is the equation of its inverse function?

Accepted Answer

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

Question 7

Which of the following is the correct formula for compound interest compounded $$n$$ times per year?

Accepted Answer

$$A(t) = P \left(1 + \frac{r}{n} \$$right)^{nt}$$

Question 8

Suppose $$P(t) = P_0$ e$$^{kt}$$ models a population with P_0$ = 1000$$ and $$k = 0.03. $$What is the doubling time $$T$$?

Accepted Answer

$$T = \frac{\ln 2}{0.03}$$

Question 9

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

Accepted Answer

$$y = \log_a(x)$$ \;\Leftrightarrow\; x = a^y$

Question 10

If $$\log_3$(x) = -2$$, what is the value of $$x$$?

Accepted Answer

$$\frac{1}{9}$$

Question 11

Which property of logarithms allows you to write $$\log_a(MN) = \log_a M + \log_a N$$?

Accepted Answer

Product Rule

Question 12

A radioactive substance has a half-life of 10 years. What is the decay constant $$k in $$the exponential decay model $$P(t) = P_0$ e$$^{-kt}$$?

Accepted Answer

$$k = \frac{\ln 2}{10}$$

Question 13

Which of the following is the correct change-of-base formula for logarithms?

Accepted Answer

$$\log_b M = \frac{\log_a M}{\log_a b}$$

Question 14

If $$f(x) = 3x + 2$$ and $$g(x) = ax + b$$ with $$g(2) = 11$$ and $$f^{-1}(15) = g(1$$)$$, what are the values of a$ and b$?

Accepted Answer

$$a = 4,\; b = 3$$

Question 15

Which of the following is the correct simplification of $$4^2$$ \cdot $$4^{-5}$$?

Accepted Answer

$$4^{-3}$$

Question 16

Suppose $$A(t) = 2000e$$^{0.065t}$$ models an investment. How long will it take to triple the investment?

Accepted Answer

$$t = \frac{\ln 3}{0.065}$$

Question 17

Which of the following is the correct application of the power rule for logarithms?

Accepted Answer

$$\log_a(M^p) = p \log_a M$$

Question 18

If $$\ln(4x + 6) - \ln(x + 5) = \ln x$$, what is the solution for $$x$$?

Accepted Answer

$$x = 2$$

Exponential and Logarithmic Functions: Inverses, Properties, and Applications

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