- Complex Numbers
- Adding and Subtracting Complex Numbers
- Conjugate of Complex Numbers
- Multiplying and Dividing Complex Numbers
- Quadratic Equations
- Solving Quadratic Equations with Complex Solutions
- Number and Nature of Solutions of Quadratic Equations
- Functions
- Evaluate a function
- Domain and Range of a function
- Operations on functions
- One-to-one functions
- Inverse of a function
- Absolute Value function
- Even and Odd functions
- Transformations of functions
- Piecewise functions
- Rate of Change of a Function
- Polynomials Functions
- Dividing Polynomials
- Factoring Polynomials
- Factoring Polynomials Using Special Polynomials
- Solving Polynomial Equations
- Solving Polynomial Inequalities
- Graphing Polynomials Functions
- Rational Expressions, Equations, Inequalities and Functions
- Simplifying Rational Expressions
- Solving Rational Equations
- Solving Rational Inequalities
- Graphing Rational Functions
- Trigonometry and Trigonometric Functions
- Converting Degrees into Radians and Vice Versa
- Evaluating Exact Values of Trigonometric Ratios
- Using Trigonometric Identities to Simplify Trigometric Expressions
- Solving Trigonometric Equations
- Graphing Trigonometric Functions
- Logarithmic and Exponential Functions
- Simplifying Exponnential Expressions
- Using Relationship Between Logarithmic and Exponential Expressions
- Simplifying Logarithmic Expressions
- Solving Logarithmic and Exponential Equations
- Graphing Logarithmic and Exponential Functions

__Complex Numbers____Quadratic Equations____Functions____Polynomials____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____Logarithmic and Exponential Functions__

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.

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.

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 \( x = a + h \).

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.

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: \( \dfrac{\cot(x)\sin(x) + \cos(x) \sin^2(x)+\cos^3(x)}{\cos(x)} \)

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) \( y = y=-log_3(x+1)-1 \)

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