# Math 105 Exam 2 Practice (Modified)

## Formulas:

$$\frac{y_2-y_1}{x_2-x_1}$$

$$y - y_1 = m(x - x_1)$$

$$P(x) = R(x) - C(x)$$

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## Question 1. Graph the equation using the slope and $y$-intercept. $y=\dfrac{3}{4}x + 1$

The equation is in $y = mx + b$ form, where $m$ is the slope, and $b$ is the $y$-intercept.
In this case, $b = 1$ and $m = \dfrac{3}{4}$. Start from the point $(0,1)$ and go right 4 up 3.


## Question 2. Find the $y = mx + b$ form for the equation of the line through $(5, -6)$ with slope $\dfrac{2}{3}$.

$x_1=5$, $y_1=-6$, and $m=\dfrac{2}{3}$.

\begin{align}
y-y_1&=m(x-x_1) \\
y-(-6)&=\dfrac{2}{3}(x-5) \\
y+6&=\dfrac{2}{3}(x-5) \\
y+6&=\dfrac{2}{3}x+ \dfrac{2}{3}(-5) \\
y+6&=\dfrac{2}{3}x- \dfrac{10}{3} \\
y&=\dfrac{2}{3}x- \dfrac{10}{3}-6 \\
y&=\dfrac{2}{3}x- \dfrac{10}{3}-\dfrac{6}{1} \\
y&=\dfrac{2}{3}x- \dfrac{10}{3}-\dfrac{18}{3} \\
y&=\dfrac{2}{3}x- \dfrac{28}{3}
\end{align}

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## Question 3. Consider the line $3x - 5y = 10$.

### a. What is its slope?

\begin{align}
3x - 5y &= 10 \\
-5y&=-3x+10 \\
\dfrac{-5y}{-5}&=\dfrac{-3x+10}{-5} \\
y&=\dfrac{-3x}{-5}+\dfrac{10}{-5} \\
y&=\dfrac{3}{5}x-2
\end{align}

The slope is $m=\dfrac{3}{5}$

### b. What is the equation of the line perpendicular to this line through $(2, -3)$?

The slope of a line perpendicular is $m=-\dfrac{5}{3}$.

\begin{align}
y-y_1&=m(x-x_1) \\
y-(-3)&=-\dfrac{5}{3}(x-2) \\
y+3&=-\dfrac{5}{3}(x) -\dfrac{5}{3}(-2) \\
y+3&=-\dfrac{5}{3}x+\dfrac{10}{3} \\
y&=-\dfrac{5}{3}x+\dfrac{10}{3}-3 \\
y&=-\dfrac{5}{3}x+\dfrac{10}{3}-\dfrac{9}{3} \\
y&=-\dfrac{5}{3}x+\dfrac{1}{3}
\end{align}

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## Question 4. Consider the line $4x - 11y = 14$.

### a. What is the slope of this line?

### b. What is the equation of the line parallel to this line through $(0, -1)$?

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## Question 5. Consider the points $(3, 6)$ and $(-9, 2)$.

### a. What is the slope through these two points?

The slope is $\dfrac{y_2-y_1}{x_2-x_1}$

\begin{align}
m&=\dfrac{y_2-y_1}{x_2-x_1} \\
&=\dfrac{2-6}{-9-3} \\
&=\dfrac{-4}{-12} \\
&=\dfrac{1}{3}.
\end{align}

### b. What is the equation of the line through these points?

$y-y_1=m(x-x_1)$

$y-6=\dfrac{1}{3}(x-3)$
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## Question 6. Find the equation of the line through $(5, -1)$ parallel to the line $x = 6$.

$x=5$

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## Question 7. Find the equation of the line through $(2, 3)$ perpendicular to the line $x = 4$.

$y=3$

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## Question 8. Solve the equation $3(2x + 4) - 5(-x + 6) = 2(3x - 7)$.

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## Question 9. Solve the equation $4(2x - 1) - 3(x + 7) = 5(x - 4)$.

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## Question 10. Solve the inequality. Write the solution in interval notation. $3 - 4(2x - 1) \leq x + 7$.

\begin{align}
3-4(2x - 1)&\leq x + 7 \\
3-12x+4 &\leq x + 7 \\
-12x+7 &\leq x + 7 \\
-13x +7&\leq +7\
-13x&\leq 0 \\
\dfrac{-13x}{-13} &\geq 0 \\
x &\geq 0
\end{align}

The solution in interval notation is $\left[0,\infty\right)$.

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## Question 11. Find the domain of $f(x) = \sqrt{4x + 7}$.

$4x+7 \geq 0$
$4x \geq -7$

$x \geq -\dfrac{7}{4}$

Interval $\left[-\dfrac{7}{4},\infty\right)$.

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## Question 12. Find the domain of $f(x) = \dfrac{1}{\sqrt{5x + 12}}$.

$5x+12 \geq 0$
$5x \geq -12$

$x \geq -\dfrac{12}{5}$

Interval $\left[-\dfrac{12}{5},\infty\right)$.

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## Question 13. A graph of a function $f(x)$ is shown. Using the graph, state the intervals where $f(x)$ is increasing and decreasing. Also, state the relative maximum and minimum values for $f(x)$.

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## Question 14. A graph of a function $f(x)$ is shown. Using the graph, state the intervals where $f(x)$ is increasing, decreasing, and constant.

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## Question 15. Let $$f(x) = \begin{cases} \dfrac{1}{2}x - 3 & \text{if } x \leq 0 \\ x^2 & \text{if } 0 < x < 4 \\ 3x + 1 & \text{if } x \geq 4 \end{cases}$$

### a. Find $f(6)$.

### b. Find $f(3)$.

### c. Find $f(-2)$.

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## Question 16. Let $f(x) = x^2 + 3$ and $g(x) = 4x - 3$. Calculate $(f+g)(3)$ and $(fg)(-2)$.

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## Question 17. Let $f(x) = x^2 + 3$ and $g(x) = 4x - 1$. Calculate $(f-g)(4)$ and $(f/g)(2)$.

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## Question 18. Let $f(x) = x - 4$ and $g(x) = 2x + 6$. Determine and simplify $(f+g)(x)$ and $(fg)(x)$.

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## Question 19. Let $f(x) = x^3$ and $g(x) = 5x - 3$. Determine and simplify $(f-g)(x)$ and $(fg)(x)$.

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## Question 20. Let $f(x) = x^4$ and $g(x) = 2x^5 - x^2 + 3$. Determine and simplify $(fg)(x)$ and $(g/f)(x)$.

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## Question 21. Let $f(x) = x^3$ and $g(x) = 4x^4 - 3x^3 + 1$. Determine and simplify $(ff)(x)$ and $(g/f)(x)$.

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## Question 22. Let $f(x) = x^5$ and $g(x) = 4x^4 - x^3 + 1$. Determine and simplify $(f-g)(x)$ and $(f/g)(x)$.

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## Question 23. If the revenue from producing $x$ units is $R(x) = 25x - 3x^2$ and the cost from producing $x$ units is $C(x) = 7x + 5$, what is the profit $P(x)$?

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