College Algebra Exam Practice: Functions, Equations, and Graphs

Graphs, Functions, and Models

Basic Operations and Properties of Functions

Understanding functions and their properties is fundamental in College Algebra. Functions can be combined, decomposed, and analyzed for their behavior and characteristics.

• Function Definition: A function is a relation that assigns each input exactly one output.

• Function Notation: $f(x)$ denotes the output of function $f$ for input $x$.

• Operations on Functions: Functions can be added, subtracted, multiplied, divided, and composed.

• Composition of Functions: $(f ext{ o } g)(x) = f(g(x))$.

• Example: If $f(x) = 2x^2$ and $g(x) = 3x^4 - 2x^2 + 2$, then $(g/f)(x) = \frac{g(x)}{f(x)}$.

Graphing and End Behavior

Graphs visually represent functions and their behavior. The end behavior describes how the function behaves as $x$ approaches infinity or negative infinity.

• Polynomial End Behavior: Determined by the leading term (highest degree term).

• Example: For $f(x) = -3x^7 - 2x^2 - x + 4$, the leading term is $-3x^7$.

• Intercepts:

  • x-intercept: Where $f(x) = 0$.

  • y-intercept: Where $x = 0$.

• Vertical Asymptotes: Values of $x$ where the function is undefined due to division by zero.

• Horizontal Asymptotes: The value the function approaches as $x$ goes to infinity.

Quadratic Functions and Equations; Inequalities

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero. Example: $x^2 - 10x - 24 = 0$

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Example: Solve $2x^2 - 2x - 5 = 0$ using the formula.

• Completing the Square: Rearrange the equation to form a perfect square trinomial.

Roots and Multiplicity

The roots (or zeros) of a polynomial are the values of $x$ for which the polynomial equals zero. The multiplicity of a root is the number of times it appears as a factor.

• Example: For $h(x) = -2x^2(x + 2)$, the roots are $x = 0$ (multiplicity 2) and $x = -2$ (multiplicity 1).

• Behavior at Roots:

  • If the root has odd multiplicity, the graph crosses the x-axis.

  • If the root has even multiplicity, the graph touches and bounces off the x-axis.

Polynomial Functions and Rational Functions

Polynomial Functions

Polynomials are expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.

• Degree: The highest exponent of the variable.

• Leading Coefficient: The coefficient of the term with the highest degree.

• End Behavior: Determined by the degree and leading coefficient.

Rational Functions

Rational functions are quotients of polynomials. Their graphs may have vertical and horizontal asymptotes.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is not zero.

• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator.

  • If degrees are equal: $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$

  • If numerator degree < denominator degree: $y = 0$

  • If numerator degree > denominator degree: No horizontal asymptote (may have an oblique asymptote).

• Example: $g(x) = \frac{3x^2 + 2}{x^2 + 5x}$

Table: Asymptote Classification

Type	Condition	Equation
Vertical Asymptote	Denominator = 0, Numerator ≠ 0	$x = a$
Horizontal Asymptote	Degrees equal	$y = \frac{a_n}{b_n}$
Horizontal Asymptote	Numerator degree < denominator degree	$y = 0$

More on Functions

Function Composition and Decomposition

Functions can be composed to form new functions, or decomposed into simpler functions.

• Composition: $(f ext{ o } g)(x) = f(g(x))$

• Decomposition: Given $h(x)$, find $f(x)$ and $g(x)$ such that $h(x) = f(g(x))$.

• Example: If $h(x) = (7 - 2x)^2$, possible decomposition: $g(x) = 7 - 2x$, $f(x) = x^2$.

Solving Equations

Linear and Quadratic Equations

Equations can be solved by isolating the variable, factoring, or using formulas.

• Linear Equation: $ax + b = 0$

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring: Express as product of factors and set each to zero.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: $x^2 - 8x + 7 = 0$

Solving Radical Equations

Equations involving roots require isolating the radical and squaring both sides.

• Example: $7 + \sqrt{2x - 5} = 10$

Additional Topics

Fractional Equations and Simplification

Solving equations with fractions involves finding a common denominator and simplifying.

• Example: $\frac{1}{4} + \frac{1}{2} = \frac{1}{3} + \frac{1}{5}$

Zeros and Multiplicity

Finding zeros and their multiplicity helps in graphing and understanding polynomial behavior.

• Example: $A(x) = (x + 1)^2(x + i)^2x^2$

Graphing Rational Functions

Graphing involves identifying intercepts and asymptotes, and sketching the general shape.

• Example: $f(x) = \frac{2x + 2}{3x - 6}$

• Steps:

  1. Find x-intercept: Set numerator to zero.

  2. Find y-intercept: Set $x = 0$.

  3. Find vertical asymptote: Set denominator to zero.

  4. Find horizontal asymptote: Compare degrees.

Table: Steps for Graphing Rational Functions

Step	Description
1	Find x-intercept
2	Find y-intercept
3	Find vertical asymptote
4	Find horizontal asymptote
5	Sketch graph

Additional info: These notes cover topics from College Algebra including functions, equations, polynomials, rational functions, graphing, and asymptotes, as reflected in the exam practice questions.