Q: A company’s profit is modeled by $P(x) = -x^2 + 6x - 4$, where $x$ is the number of units sold. What is the maximum profit and at how many units sold does it occur?

Maximum profit is $5$ at $x = 3$ units.

Question 1

Given the function $f(x) = 5x + 8$, what is the equation for its inverse function $f^{-1}(x)$?

Accepted Answer

$f^{-1}(x) = \frac{x - 8}{5}$

Question 2

Find the inverse of the function $f(x) = \frac{3x + 4}{2x + 1}$.

Accepted Answer

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

Question 3

For the quadratic function $f(x) = -3(x - 2)^2 + 5$, what are the coordinates of the vertex?

Accepted Answer

$(2, 5)$

Question 4

Given $f(x) = -x^2 + 6x - 4$, does the function have a maximum or minimum value, and what is the value?

Accepted Answer

Maximum at $x = 3$, value $f(3) = 5$

Question 5

What is the degree of the polynomial $x^3 + 2x^2 - 5x - 6$?

Accepted Answer

3

Question 6

What is the leading coefficient of the polynomial $x^3 + 2x^2 - 5x - 6$?

Accepted Answer

1

Question 7

Find the $x$-intercepts of the polynomial $x^3 + 2x^2 - 5x - 6$.

Accepted Answer

$x = -3, x = -1, x = 2$

Question 8

Divide $x^2 + 2x^2 - 5x - 6$ by $x + 1$ using synthetic division. What is the quotient?

Question 9

Write a third-degree polynomial function with real coefficients that has zeros at $x = -3$, $x = 1$, and $x = 5$.

Accepted Answer

$f(x) = (x + 3)(x - 1)(x - 5)$

Question 10

Find the vertical asymptotes of $f(x) = \frac{x+4}{x-2}$.

Accepted Answer

$x = 2$

Question 11

Find the vertical asymptotes of $f(x) = \frac{x^2+3x}{x^2-9}$.

Accepted Answer

$x = 3$ and $x = -3$

Question 12

Find the vertical asymptotes of $f(x) = \frac{x^2-5}{x^2-3}$.

Accepted Answer

$x = \sqrt{3}$ and $x = -\sqrt{3}$

Question 13

Find the horizontal asymptote of $f(x) = \frac{2x^2+4}{x^2-1}$.

Accepted Answer

$y = 2$

Question 14

Find the horizontal asymptote of $f(x) = \frac{5x^2+3}{2x^2+1}$.

Accepted Answer

$y = \frac{5}{2}$

Question 15

Find the horizontal asymptote of $f(x) = \frac{4x^2+1}{4x^2+3}$.

Accepted Answer

$y = 1$

Question 16

Which of the following could be the graph of $f(x) = x^3 - 4x^2$?

Accepted Answer

A cubic graph with $x$-intercepts at $x = 0$ and $x = 4$, and a local minimum between them.

Question 17

A chemical reaction rate is modeled by $f(x) = \frac{3x+4}{2x+1}$. If the rate must not exceed 2, what is the maximum value of $x$ allowed?

Accepted Answer

$x < 1$

Question 18

A company’s profit is modeled by $P(x) = -x^2 + 6x - 4$, where $x$ is the number of units sold. What is the maximum profit and at how many units sold does it occur?

Accepted Answer

Maximum profit is $5$ at $x = 3$ units.

Question 19

A medical dosage is modeled by $f(x) = 5x + 8$. If the maximum safe dosage is 33 mg, what is the maximum value of $x$ that can be administered?

Accepted Answer

$x = 5$

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