Complex Numbers
Problem 1-1
Let z = 2 - 3 i where i is the imaginary unit. Evaluate z z* , where z* is the conjugate of z , and write the answer in standard form.
Problem 1-2
Evaluate and write in standard form \( \dfrac{1-i}{2-i} \) , where i is the imaginary unit.
Quadratic Equations
Problem 2-1
Find all solutions of the equation \( x(x + 3) = - 5 \).
Problem 2-2
Find all values of the parameter m for which the equation \( -2 x^2 + m x = 2 m \) has complex solutions.
Functions
Problem 3-1
Let \( f(x) = - x^2 + 3(x - 1) \). Evaluate and simplify \( f(a-1)\).
Problem 3-2
Write, in interval notation, the domain of function \(f\) given by \(f(x) = \sqrt{x^2-16} \).
Problem 3-3
Find and write, in interval notation, the range of function \(f\) given by \(f(x) = - x^2 - 2x + 6 \).
Problem 3-4
Let \(f(x) = \sqrt{x - 2} \) and \(g(x) = x^2 + 2 \); evaluate \( (f_o g)(a - 1) \) for \( a \lt 1 \).
Problem 3-5
Which of the following is a one-to-one function?(There may be more than one answer).
a) \(f(x) = - 2 \) b) \(g(x) = \ln(x^2 - 1) \) c) \(h(x) = |x| + 2 \) d) \(j(x) = 1/x + 2 \) e) \(k(x) = \sin(x) + 2 \) f) \(l(x) = ln(x - 1) + 1 \)
Problem 3-6
What is the inverse of function f given by \(f(x) = \dfrac{-x+2}{x-1}\)?
Problem 3-7
Classify the following functions as even, odd or neither.
a) \(f(x) = - x^3 \) b) \(g(x) = |x|+ 2 \) c) h(x) = \( \ln(x - 1) \)
Problem 3-8
Function \(f \) has one zero only at \(x = -2\). What is the zero of the function \(2f(2x - 5) \)?
Problem 3-9
Which of the following piecewise functions has the graph shown below?
a) \( f(x) = \begin{cases}
x^2 & \text{if} \; x \ge 0 \\
2 & \text{if} \; -2 \lt x \lt 0\\
- x + 1& \text{if} \; x \le -2
\end{cases} \)
b) \( g(x) = \begin{cases}
x^2 & \text{if} \; x \gt 0 \\
2 & \text{if} \; -2 \lt x \le 0\\
- x + 1& \text{if} \; x \le -2
\end{cases} \)
c) \( h(x) = \begin{cases}
x^2 & \text{if} \; x \gt 0 \\
2 & \text{if} \; -2 \lt x \lt 0\\
- x + 1 & \text{if} \; x \lt -2
\end{cases} \)
Problem 3-10
Calculate the average rate of change of function \( f(x) = \dfrac{1}{x} \) as x changes from \( x = a\) to \( x = a + h \).
Polynomials
Problem 4-1
Find the quotient and the remainder of the division \( \dfrac{-x^4+2x^3-x^2+5}{x^2-2} \).
Problem 4-2
Find \( k \) so that the remainder of the division \( \dfrac{4 x^2+2x-3}{2 x + k} \) is equal to \( -1 \)?
Problem 4-3
\( (x - 2) \) is one of the factors of \( p(x) = -2x^4-8x^3+2x^2+32x+24 \). Factor \(p\) completely.
Problem 4-4
Factor \( 16 x^4 - 81 \) completely.
Problem 4-5
Find all solutions to the equation \( (x - 3)(x^2 - 4) = (- x + 3)(x^2 + 2x) \)
Problem 4-6
Solve the inequality \( (x + 2)(x^2-4x-5) \ge (-x - 2)(x+1)(x-3)\)
Problem 4-7
The graph of a polynomial function is shown below. Which of the following functions can possibly have this graph?
a) \( y = -(x+2)^5(x-1)^2 \) b) \( y = 0.5(x+2)^3(x-1)^2 \) c) \( y = -0.5(x+2)^3 (x-1)^2 \) d) \( y = -(x+2)^3(x-1)^2 \)
Problem 4-8
Which of the following graphs could possibly be that of the function f given by \( f(x) = k (x - 1)(x^2 + 4) \) where k is a negative constant? Find k if possible.
Rational Expressions, Equations, Inequalities and Functions
Problem 5-1
Write as a single rational expression: \( \dfrac{x^2+3x-5}{(x-1)(x+2)} - \dfrac{2}{x+2} - 1 \).
Problem 5-2
Solve the equation: \( \dfrac{- x^2+5}{x-1} = \dfrac{x-2}{x+2} - 4 \).
Problem 5-3
Solve the inequality: \( \dfrac{1}{x-1}+\dfrac{1}{x+1} \ge \dfrac{3}{x^2-1} \).
Problem 5-4
Find the horizontal and vertical asymptotes of the function: \( y = \dfrac{3x^2}{5 x^2 - 2 x - 7} + 2 \).
Problem 5-5
Which of the following rational functions has an oblique asymptote? Find the point of intersection of the oblique asymptote with the function.
a) \( y = -\dfrac{x-1}{x^2+2} \) b) \( y = -\dfrac{x^4-1}{x^2+2} \) c) \( y = -\dfrac{x^3 + 2x ^ 2 -1}{x^2- 2} \) d) \( y = -\dfrac{x^2-1}{x^2+2} \)
Problem 5-6
Which of the following graphs could be that of function \( f(x) = \dfrac{2x-2}{x-1} \)?
Trigonometry and Trigonometric Functions
Problem 6-1
A rotating wheel completes 1000 rotations per minute. Determine the angular speed of the wheel in radians per second.
Problem 6-2
Determine the exact value of \( sec(-11\pi/3) \).
Problem 6-3
Convert 1200° in radians giving the exact value.
Problem 6-4
Convert \( \dfrac{-7\pi}{9} \) in degrees giving the exact value.
Problem 6-5
What is the range and the period of the the function \( f(x) = -2\sin(-0.5(x - \pi/5)) - 6 \)?
Problem 6-6
Which of the following graphs could be that of function given by: \( y = - \cos(2x - \pi/4) + 2 \)?
Problem 6-7
Find a possible equation of the form \( y = a \sin(b x + c) + d \) for the graph shown below.(there are many possible solutions)
Problem 6-8
Find the smallest positive value of x, in radians, such that \( - 4 \cos (2x - \pi/4) + 1 = 3 \)
Problem 6-9
Simplify the expression: \( \dfrac{\cot(x)\sin(x) + \cos(x) \sin^2(x)+\cos^3(x)}{\cos(x)} \)
Logarithmic and Exponential Functions
Problem 7-1
Simplify the expression \( \dfrac{4x^2 y^8}{8 x^3 y^5} \) using positive exponents in the final answer.
Problem 7-2
Evaluate the expression \( \dfrac{3^{1/3} 9^{1/3}}{4^{1/2}} \).
Problem 7-3
Rewrite the expression \( \log_b(2x - 4) = c \) in exponential form.
Problem 7-4
Simplify the expressiomn: \( \log_a(9) \cdot \log_3(a^2) \)
Problem 7-5
Solve the equation \( \log(x + 1) - log(x - 1) = 2 \log(x + 1) \).
Problem 7-6
Solve the equation \( e^{2x} + e^x = 6 \).
Problem 7-7
What is the horizontal asymptote of the graph of \( f(x) = 2 ( - 2 - e^{x-1}) \)?
Problem 7-8
What is the vertical asymptote of the graph of \( f(x) = log(2x - 6) + 3 \)?
Problem 7-9
Match the given functions with the graph shown below?
A) \( y = 2 - 0.5^{2x-1} \) B) \( y = 0.5^{2x-1} \) C) \( y = 2 - 0.5^{-2x+1} \) D) \( y = 0.5^{-2x+1} \)
Problem 7-10
Match the given functions with the graph shown below?
A) \( y = 2+ln(x-2) \) B) \( y=-log_2(x+1)-1 \) C) \( y = -ln(-x) \) D) \( y = y=-log_3(x+1)-1 \)
