Q: A car’s value $V$ depreciates exponentially according to $V(t) = 20000 \times 0.85^t$, where $t$ is years after purchase. What is the value after 5 years?

$V(5) = 20000 \times 0.85^5 \approx 8867$

Question 1

Which of the following is the correct definition of a rational function?

Accepted Answer

A function that can be expressed as the ratio of two polynomials, where the denominator is not zero.

Question 2

Given the quadratic function $f(x) = 2x^2 - 4x + 1$, what is the vertex of the parabola?

Accepted Answer

$(1, -1)$

Question 3

A chemical reaction's rate $R$ is modeled by the exponential function $R(t) = 5e^{-0.2t}$, where $t$ is time in minutes. What is the rate after 10 minutes?

Accepted Answer

$R(10) = 5e^{-2} \approx 0.68$

Question 4

Which of the following is the correct solution set for the inequality $x^2 - 5x + 6 < 0$?

Accepted Answer

$2 < x < 3$

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

$x 
eq 4$

Question 7

A student is analyzing the graph of $y = \log_2(x-1)$. Which of the following statements is true about its domain?

Accepted Answer

$x > 1$

Question 8

A quadratic equation models the height $h$ (in meters) of a ball thrown upward: $h(t) = -4.9t^2 + 20t + 2$. What is the maximum height reached by the ball?

Accepted Answer

$24.4$ meters

Question 9

Which of the following is the correct solution to the equation $2^{x} = 16$?

Accepted Answer

$x = 4$

Question 10

A medical test result is modeled by the function $f(x) = 3x^2 - 12x + 15$. For what value of $x$ does the minimum occur?

Accepted Answer

$x = 2$

Question 11

Which of the following is the correct inverse of the function $f(x) = 2x + 5$?

Accepted Answer

$f^{-1}(x) = \frac{x-5}{2}$

Question 12

A company’s profit $P$ in thousands of dollars is modeled by $P(x) = -2x^2 + 12x - 16$, where $x$ is the number of units sold (in thousands). What is the maximum profit?

Accepted Answer

$P = 32$ thousand dollars

Question 13

Which of the following is the correct solution to $\log_3(x) = 4$?

Accepted Answer

$x = 81$

Question 14

A quadratic equation $ax^2 + bx + c = 0$ has real solutions if and only if:

Accepted Answer

$b^2 - 4ac \geq 0$

Question 15

A radioactive substance decays according to $A(t) = 100e^{-0.05t}$. How much remains after 20 years?

Accepted Answer

$A(20) = 100e^{-1} \approx 36.79$

Question 16

Which of the following is the correct solution to the system of equations: $\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$?

Accepted Answer

$(x, y) = (3, 1)$

Question 17

Which of the following is the correct solution to $|2x - 5| = 9$?

Accepted Answer

$x = 7$ or $x = -2$

Question 18

A car’s value $V$ depreciates exponentially according to $V(t) = 20000 	imes 0.85^t$, where $t$ is years after purchase. What is the value after 5 years?

Accepted Answer

$V(5) = 20000 	imes 0.85^5 \approx 8867$

Question 19

Which of the following is the correct solution to $x^2 - 4x + 4 = 0$?

Accepted Answer

$x = 2$ (double root)

College Algebra: Course Overview and Study Guide

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