College Algebra Study Guide: Functions, Polynomials, Rational Functions, and Graphs

Functions and Their Properties

Function Types and Definitions

Understanding the basic types and properties of functions is essential in College Algebra. Functions can be classified as even, odd, or neither, and their domains and ranges are fundamental concepts.

• Even Function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain.

• Odd Function: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain.

• Domain: The set of all possible input values ($x$) for which the function is defined.

• Range: The set of all possible output values ($f(x)$) the function can produce.

Example: The reciprocal function $f(x) = \frac{1}{x}$ is neither even nor odd because $f(-x) = -\frac{1}{x}$, which is not equal to $f(x)$ or $-f(x)$ for all $x$.

Function Operations

Functions can be combined using addition, subtraction, multiplication, and division. The composition of functions involves substituting one function into another.

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \cdot g(x)$

• Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = f(g(x)) = (x+1)^2$.

Polynomials

Polynomial Functions

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

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

• Degree: The highest exponent of $x$ in the polynomial.

• Roots/Zeros: Values of $x$ for which $f(x) = 0$.

Example: For $f(x) = 5x^2 - 3(x+2)^2$, expand and solve for roots.

Factor Theorem

The Factor Theorem states that $x - c$ is a factor of $f(x)$ if and only if $f(c) = 0$.

• Application: To find all solutions to $f(x) = 0$ when a factor is known.

Example: If $f(x) = 2x^3 + x^2 - 13x + 6$ and $x + 3$ is a factor, then $f(-3) = 0$.

Polynomial Division

Polynomial division is used to divide one polynomial by another, often to simplify rational expressions or find roots.

• Long Division: Similar to numerical long division, but with polynomials.

• Synthetic Division: A shortcut method for dividing by linear factors.

Example: Divide $x^4 - 5x^2 + 4 = 0$ given that 2 and -2 are roots.

Graphing Functions

Vertex and Range of Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$. The vertex is the highest or lowest point on the graph, and the range depends on the direction the parabola opens.

• Vertex Formula: For $f(x) = ax^2 + bx + c$, vertex at $x = -\frac{b}{2a}$

• Range: If $a > 0$, range is $[k, \infty)$; if $a < 0$, range is $(-\infty, k]$, where $k$ is the $y$-coordinate of the vertex.

Example: For $f(x) = 2(x+3)^2 - 8$, vertex is at $(-3, -8)$.

Graphing Rational Functions

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

• Vertical Asymptote: Occurs where the denominator is zero and the numerator is not zero.

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

• Hole: Occurs where both numerator and denominator are zero for the same $x$ value.

Example: For $r(x) = \frac{3x}{x^2 - 4}$, vertical asymptotes at $x = 2$ and $x = -2$.

Symmetry of Functions

Even, Odd, and Neither

Determining the symmetry of a function helps in graphing and understanding its properties.

• Even: Symmetric about the $y$-axis.

• Odd: Symmetric about the origin.

• Neither: No symmetry.

Example: $g(x) = |x| + x^2$ is even because both $|x|$ and $x^2$ are even functions.

Roots and Multiplicity

Finding Roots and Their Multiplicity

The roots of a polynomial are the values of $x$ where the function equals zero. The multiplicity of a root is the number of times it appears as a solution.

• Multiplicity: If $(x - r)^k$ is a factor, $r$ is a root of multiplicity $k$.

• Behavior at Roots: If multiplicity is odd, the graph crosses the $x$-axis; if even, it touches and turns around.

Example: For $f(x) = (x-2)^2(x+1)$, $x=2$ has multiplicity 2, $x=-1$ has multiplicity 1.

Rational Functions: Holes and Asymptotes

Identifying Holes and Asymptotes

Rational functions may have holes and asymptotes, which are important for graphing and understanding their behavior.

Function	Hole	Vertical Asymptote	Horizontal Asymptote	Slant Asymptote
$f(x) = \frac{2x^2 - 5x - 3}{x^2 + x - 12}$	At $x$ where numerator and denominator both zero (Additional info: $x=3$)	At $x$ where denominator zero and numerator not zero (Additional info: $x=-4$)	Depends on degree (Additional info: $y=2$)	None (degrees equal)
$g(x) = \frac{x^3}{x^2 - 1}$	None	$x=1$, $x=-1$	None (degree numerator > denominator)	Slant asymptote exists (Additional info: $y=x$)
$h(x) = \frac{x+1}{x}$	None	$x=0$	$y=1$	None

Additional info: Table entries inferred based on standard rational function analysis.

Applications of Polynomials

Modeling with Polynomial Functions

Polynomials can be used to model real-world phenomena, such as the number of thefts over time. However, the usefulness of a polynomial model depends on the context and the time period considered.

• Example: $f(x) = -0.87x^3 + 0.35x^2 + 81.62x + 7684.94$ models thefts in thousands in the US after 1987.

• Application: Consider whether the model remains accurate over an extended period, as higher-degree polynomials may not reflect long-term trends.

Summary Table: Key Properties of Functions

Function	Vertex	Range	Solution Set to $f(x)=0$
$f(x) = 2(x+3)^2 - 8$	$(-3, -8)$	$[-8, \infty)$	$x = -3 \pm \sqrt{4}$
$f(x) = -x^2 + 4x + 5$	$(2, 9)$	$(-\infty, 9]$	$x = 5, x = -1$
$f(x) = \frac{1}{2}x^2 - 8$	$(0, -8)$	$[-8, \infty)$	$x = \pm 4$

Additional info: Table entries inferred based on standard quadratic analysis.