**MIDTERM EXAM ON PRE-CALCULUS ALGEBRA**
**Course code: MTH1112 – Duration: 90 minutes**
**Date: November 22nd, 2019**
**Question 1.**
Consider the function
$$
f(x)=
\begin{cases}
3x + 2, & \text{if } x < -2,\\
-2x + 3, & \text{if } x \ge -2.
\end{cases}
$$
a. Graph the function
b. List the intercepts
**Question 2.**
Consider the function $f(x) = x^2 - 5x + 2$,
a. Determine whether function $f$ is even, odd or neither.
b. Find the average rate of change of $f$ from $x_0$ to $x_0 + h$, where $h$ is real.
**Question 3.**
Given the function $f(x) = (x + 2)^2 - 1$. Graph this quadratic function using the transformation.
**Question 4.**
Funfest Football Company has determined that the marginal cost C dollars of manufacturing x football clubs may be expressed by the quadratic function $C(x) = 5.3x^2 - 1081.2x + 56,100$
a. How many clubs should be manufactured to minimize the marginal cost?
b. At this level of production, what is the marginal cost?
**Question 5.**
a) Show that the following polynomial function has at least one zero in the given interval: $f(x) = 2x^4 - 3x^3 + 12x^2 - 7x - 1$ ; $[0; 1]$
b) Find any horizontal, vertical, or oblique asymptotes of the following function
$$f(x) = \frac{x^3 + 4x^2 - 3x - 5}{x^2 - 3x + 2}$$
**Question 6.**
Fourth-degree polynomial with real coefficients with zeros: $1, -2, 3 + 2i$. Find the remaining zero and write a polynomial meets the above conditions.
**Question 7.**
Find the inverse of each function and check your answer
a) $f(x) = \frac{5x+2}{-4x+3}$
b) $f(x) = 3 - \sqrt[3]{4x + 3}$
**Question 8.**
a) Write the following expression as the sum and/or difference of logarithms
$$\log_5 \frac{3x^2 + 6x + 3}{\sqrt[5]{(x^2 + 1)^4}}, x > -1$$
b) Given $\log_{12} 18 = a$; $\log_{24} 54 = b$. Prove that $ab + 5(a - b) = 1$
**Question 9.**
Solve the following equations
a) $\log_2 (3x - 1) + \log_2 (10 - 2x) = 4$
b) $(3 - 2\sqrt{2})^{x^2 - 4x} = (3 + 2\sqrt{2})^{6-x}$
**Question 10.**
a) If \$10,000 is invested at 8% compounded monthly, how much is there after 18 months?
b) If you want to have \$12,000 in 15 months, how much do you need to place in a savings account now that pays 10% compounded quarterly.
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