College Algebra Final Exam Review: Functions, Equations, Polynomials, and Applications

Functions and Their Properties

Definition of a Function

A function is a relation in which each element of the domain (input set) is paired with exactly one element of the range (output set).

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually f(x) or y-values).

• Example: If a mapping pairs 6, 7, and 3 in the domain to -6, -7, and -3 in the range, check if each input has only one output to determine if it is a function.

Evaluating Functions

To evaluate a function, substitute the given value into the function expression.

• Example: For $f(x) = x^2 + 1$, find $f(2)$ and $f(a)$:

• $f(2) = 2^2 + 1 = 5$

• $f(a) = a^2 + 1$

Domain and Range in Interval Notation

Express the set of possible x-values (domain) and y-values (range) using interval notation.

• Example: For $g(x) = \sqrt{x-1}$, the domain is $[1, \infty)$ because the expression under the square root must be non-negative.

Polynomials and Factoring

Multiplying Polynomials

Use the distributive property (FOIL for binomials) to expand products of polynomials.

• Example: $(x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$

Factoring Polynomials

Factoring expresses a polynomial as a product of its factors.

• Example: $x^2 - 10x + 25 = (x-5)^2$

• Common methods: factoring trinomials, difference of squares, grouping.

Rational Expressions and Operations

Simplifying Rational Expressions

Reduce rational expressions by factoring numerators and denominators and canceling common factors.

• Example: $\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1$ (for $x \neq 1$)

Adding, Subtracting, Multiplying, and Dividing Rational Expressions

• Find a common denominator for addition and subtraction.

• Multiply numerators and denominators directly for multiplication.

• For division, multiply by the reciprocal of the divisor.

Radical Expressions and Rational Exponents

Simplifying Radical Expressions

Express radicals in simplest form and use properties of exponents to simplify.

• Example: $\sqrt{48x^7} = 4x^3\sqrt{3x}$

Rationalizing the Denominator

To rationalize, multiply numerator and denominator by a suitable radical to eliminate radicals from the denominator.

• Example: $\frac{8\sqrt{7}}{\sqrt{2}} = \frac{8\sqrt{7} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{8\sqrt{14}}{2} = 4\sqrt{14}$

Expressing with Rational Exponents

Recall that $a^{m/n} = \sqrt[n]{a^m}$.

• Example: $\sqrt[3]{x^2} = x^{2/3}$

Solving Equations and Inequalities

Linear and Quadratic Equations

• Solve by isolating the variable, factoring, or using the quadratic formula:

$ax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: $x^2 - 4x + 3 = 0 \implies (x-1)(x-3) = 0 \implies x = 1, 3$

Solving Rational and Radical Equations

• For rational equations, multiply both sides by the least common denominator (LCD).

• For radical equations, isolate the radical and raise both sides to the appropriate power.

Solving Inequalities

• Test intervals between critical points (where the expression is zero or undefined).

• Express solutions in interval notation.

• Example: $x^2 + 21 > 10x$

Functions: Graphs and Applications

Graphing Functions

• Identify key features: intercepts, vertex (for quadratics), asymptotes (for rational functions), intervals of increase/decrease.

• Use the graph to answer questions about function values, domain, range, and behavior.

Applications of Functions

• Word problems may involve distance, rate, time, or geometric applications.

• Example: If a toy rocket's height is given by $h(t) = -16t^2 + 48t + 8$, find the maximum height by finding the vertex of the parabola.

• Vertex formula: $t = -\frac{b}{2a}$ for $ax^2 + bx + c$

Quadratic Functions and Parabolas

Vertex and Axis of Symmetry

• The vertex of $f(x) = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$.

• The axis of symmetry is the vertical line $x = -\frac{b}{2a}$.

Intercepts

• x-intercepts: Set $f(x) = 0$ and solve for $x$.

• y-intercept: Evaluate $f(0)$.

Polynomial and Rational Functions: Zeros and Asymptotes

Finding Zeros of Polynomials

• Set the polynomial equal to zero and solve for $x$ by factoring or using the Rational Root Theorem.

Asymptotes of Rational Functions

• Vertical asymptotes: Values of $x$ that make the denominator zero (and do not cancel with the numerator).

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

• Example Table:

Function	Vertical Asymptote	Horizontal Asymptote
$f(x) = \frac{1}{x}$	$x=0$	$y=0$
$f(x) = \frac{2x}{x-1}$	$x=1$	$y=2$
$f(x) = \frac{x^2}{x^2-1}$	$x=1, -1$	$y=1$

Summary Table: Key Concepts

Topic	Key Formula/Fact	Example
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$	$x^2-4x+3=0$
Vertex of Parabola	$x = -\frac{b}{2a}$	$f(x) = 2x^2-8x+3$
Rational Exponents	$a^{m/n} = \sqrt[n]{a^m}$	$8^{2/3} = (\sqrt[3]{8})^2 = 4$
Domain of $\sqrt{x-a}$	$x \geq a$	$\sqrt{x-3}$: $x \geq 3$

Additional info: This review covers core College Algebra topics including functions, polynomials, rational and radical expressions, equations, inequalities, graphing, and applications. Students should practice each type of problem and understand the underlying concepts for exam success.