Final Exam

Solving Linear Equations

• Solve:

  • Equation: 2x - 1 = 1

Solving Square Root Equations

• Solve:

  • Equation: \sqrt{2x - 3} = 3 - x

Writing Equations of Lines

• Write the equation of the line passing through the points (1,2) and (3,4).

Perimeter and Area Problems

• Problem Statement: The perimeter of a rectangular garden is 64 ft. The length is 4 ft longer than the width.

  • Objective: Find the dimensions of the garden.

Expressing Numbers in Complex Form

• Write -18 in terms of i:

  • Simplify:

Multiplying Complex Numbers

• Simplify in complex (standard) form:

  • Expression: (1 - i)(5 + i)

Basic Arithmetic Operations

• Solve:

  • Expression: 64 - 2 = 0

Solving Quadratic Equations

• Solve:

  • Equation: 3x^2 - 2x - 1 = 0

• Solve using the quadratic formula:

  • Equation: x^2 + 2x - 1 = 0

Solving Inequalities

• Solve the inequality:

  • \frac{x - 2}{x + 4} \leq 0

• Solve the inequality:

  • |13x + 12| \geq 9

Solving Polynomial Equations

• Solve:

  • Equation: x^4 - 2x^2 + 1 = 0

Finding Domain of Functions

• Write the domain in interval notation for:

  • Function: f(x) = \sqrt{\sqrt{x - 5}}

Transformations and Graphing Functions

• Consider the function: f(x) = -2|x| + 2

  • a) List the transformations from the basic graph |x| in the correct order.

  • b) Sketch the graph on your scratch paper, labeling your work step by step:

  • Note: Credit will not be given if you only give the final graph.

Function Analysis

• Determine if the function: f(x) = -x^5 + x^3 is even, odd, or neither.

  • Objective: Show your work supporting your reasoning.

Evaluating Piecewise Functions

• Given the piecewise function:

  • f(x) = \begin{cases} 1 - 6, & x \leq 0 \ x^2, & x > 0 \end{cases}

  • Evaluate:

  • f(2)

Finding Zeros and Their Multiplicities

• Determine the zeros & multiplicities for the function:

  • Function: f(x) = (x + 2)(x - 1)^2(x - 4)^3

  • Results:

  • x = -2, ext{ multiplicity } 1

  • x = 1, ext{ multiplicity } 2

  • x = 4, ext{ multiplicity } 3

Finding Inverses of Functions

• Find the inverse of the following one-to-one function:

  • Function: f(x) = 3x - 2

  • Inverse:

  • f^{-1}(x) = \frac{x + 2}{3}

Composition of Functions

• Given the functions:

  • f(x) = 4x + 3

  • g(x) = 2x^2 + 1

  • Find the composition function:

  • (g \circ f)(x)

  • Result: 32x^2 + 48x + 19

Function Operation

• Given the functions:

  • f(x) = 2x + 1

  • g(x) = x - 1

  • Find:

  • (f-g)(x)

  • Result: x + 2

Polynomial Division

• Divide:

  • Expression: (2x^2 + 3x - 14) + (x - 1)

Remainder Theorem and Zeros of Polynomials

• Using the Remainder Theorem, determine:

  • Statement: 0 is a zero of the polynomial f(x) = -2x + 4x^3 + 18 + x^4

Identifying Rational Zeros

• List all possible rational zeros of:

  • f(x) = 7x^4 + 5x^3 - 7x^2 + 14

  • Correct List: \pm 1, \pm 2, \pm 7, \pm 14

Using the Factor Theorem

• Using Factor Theorem, determine:

  • Statement: (x - 2) is a factor of f(x) = 2x^3 - 3x^2 - 5x + 6

Graphing Exponential Functions

• Match the graph with its exponential function:

  • X-axis Range: (-5, 5)

  • Y-axis Range: [0, 4]

Finding Asymptotes of Rational Functions

• Find all vertical asymptotes (VA) and horizontal asymptotes (HA) of the rational function:

  • Function: f(x) = \frac{x - 3}{x^2 - 1}

Finding Zeros with a Given Zero

• Find the zeros of:

  • f(x) = x^3 - 7x^2 + 7x + 15

  • Given: 5 is a zero.

  • Correct Zeros: {-1, 3, 5}