Q: A bank account earns $10.25\%$ interest compounded continuously. How much should you invest now to have $100,000$ in $18$ years?

$P = \frac{100000}{e^{0.1025 \times 18}}$

Question 1

Which of the following statements correctly describes the relationship between a function $f$ and its inverse $f^{-1}$?

Accepted Answer

For all $x$ in the domain of $f$, $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Question 2

Given the function $f(x) = \sqrt{x+1}$, what is the inverse function $f^{-1}(x)$?

Question 3

Suppose the temperature $F$ in degrees Fahrenheit is given by $F = \frac{9}{5}C + 32$, where $C$ is the temperature in degrees Centigrade. What is the inverse function that gives $C$ in terms of $F$?

Accepted Answer

$C = \frac{5}{9}(F - 32)$

Question 4

Which of the following functions is self-inverse (i.e., its own inverse)?

Accepted Answer

$g(x) = \frac{6}{x}$

Question 5

A function $f$ is one-to-one if and only if:

Accepted Answer

Each output value is produced by exactly one input value.

Question 6

Given the table below, what is $f^{-1}(4)$?

$x$: $-4$, $-3$, $1$, $2$
$f(x)$: $3$, $1$, $-3$, $4$

Accepted Answer

$f^{-1}(4) = 2$

Question 7

Which graphical test can be used to determine if a function has an inverse?

Accepted Answer

Horizontal line test

Question 8

If $f(x) = |x - 7|$, does $f$ have an inverse over its entire domain?

Accepted Answer

No, because $f$ is not one-to-one.

Question 9

Given the exponential function $f(x) = 10 \cdot 2^x$, what is the initial value and the base?

Accepted Answer

Initial value: $10$, Base: $2$

Question 10

A population of wild rabbits doubles every year. If there are initially $100$ rabbits, which equation models the population after $x$ years?

Accepted Answer

$y = 100 \cdot 2^x$

Question 11

Given the compound interest formula $A = P \left(1 + \frac{r}{n}\right)^{nt}$, what does $n$ represent?

Accepted Answer

The number of times interest is compounded per year

Question 12

If $P = 1500$, $r = 0.065$, $n = 4$, and $t = 5$, what is the value of $A$ using the compound interest formula?

Accepted Answer

$A = 1500 \left(1 + \frac{0.065}{4}\right)^{20}$

Question 13

What is the value of $e^{1.8}$ to three decimal places?

Accepted Answer

$e^{1.8} \approx 6.050$

Question 14

If interest is compounded continuously, which formula should be used?

Accepted Answer

$A = P e^{rt}$

Question 15

A bank account earns $10.25\%$ interest compounded continuously. How much should you invest now to have $100,000$ in $18$ years?

Accepted Answer

$P = \frac{100000}{e^{0.1025 	imes 18}}$

Question 16

Which of the following is the correct inverse of $f(x) = 3x - 8$?

Accepted Answer

$f^{-1}(x) = \frac{x+8}{3}$

Question 17

Which of the following statements about the graph of a function and its inverse is true?

Accepted Answer

The graph of the inverse is a reflection of the graph of the function across the line $y = x$

Question 18

If the horizontal asymptote of a function is $y = -2$, what does it become for the inverse function?

Accepted Answer

A vertical asymptote at $x = -2$

Question 19

Which of the following functions is NOT one-to-one?

Accepted Answer

$h(x) = -\frac{1}{4}(x+1)^2$

Inverse Functions and Exponential Functions in College Algebra

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