Question 1

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

Accepted Answer

A function where each value in the domain corresponds to a unique value in the range, and each value in the range corresponds to a unique value in the domain.

Question 2

Given the function $f(x) = 2x + 7$, which of the following is the inverse function $f^{-1}(x)$?

Accepted Answer

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

Question 3

Which test can be used to determine if a function is one-to-one by inspecting its graph?

Accepted Answer

Horizontal Line Test

Question 4

If $f(x) = x^2$ for $x \in \mathbb{R}$, is $f$ a one-to-one function?

Accepted Answer

No, because $f(-x) = f(x)$ for all $x$.

Question 5

Suppose $f(x) = \sqrt{x-5}$, what is the domain of $f^{-1}(x)$?

Accepted Answer

$x \geq 0$

Question 6

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

Accepted Answer

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

Question 7

The graph of an inverse function $f^{-1}$ is a reflection of the graph of $f$ over which line?

Accepted Answer

$y = x$

Question 8

Which of the following is the correct domain and range for the exponential function $f(x) = e^x$?

Accepted Answer

Domain: $(-\infty, \infty)$, Range: $(0, \infty)$

Question 9

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

Accepted Answer

$P(t) = 500 \cdot 2^{t/3}$

Question 10

If $A = P\left(1 + \frac{r}{n}\right)^{nt}$, what does the variable $n$ represent in the context of compound interest?

Accepted Answer

The number of times interest is compounded per year

Question 11

Which of the following is the correct logarithmic form of the exponential equation $2^x = 16$?

Accepted Answer

$\log_2 16 = x$

Question 12

What is the value of $\log_{10} 1000$?

Accepted Answer

$3$

Question 13

Which property of logarithms allows you to write $\log_b(xy)$ as $\log_b x + \log_b y$?

Accepted Answer

Product Property

Question 14

Solve for $x$: $5^{2x-1} = 125$

Accepted Answer

$x = 2$

Question 15

Which of the following is the correct change-of-base formula for $\log_a b$?

Accepted Answer

$\log_a b = \frac{\log_c b}{\log_c a}$

Question 16

If $A = 5000e^{0.05t}$ models the value of an investment with continuous compounding, what is the value after 10 years?

Accepted Answer

$A = 5000e^{0.5}$

Question 17

Which of the following best describes the range of the logarithmic function $f(x) = \log_b x$ for $b > 1$?

Accepted Answer

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

Question 18

A radioactive substance has a half-life of 4 years. If the initial amount is 800 grams, how much remains after 8 years?

Accepted Answer

$200$ grams

Question 19

Which of the following is the correct exponential model for carbon dating, where $A$ is the amount present after $t$ years, $A_0$ is the initial amount, and $k$ is the decay constant?

Accepted Answer

$A = A_0 e^{-kt}$

Inverse, Exponential, and Logarithmic Functions: College Algebra Study Notes

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