College Algebra: Functions and Polynomial Fundamentals

Functions and Their Properties

Definition of a Function

A function is a relation that assigns exactly one output value to each input value. In mathematical terms, a function f from set A to set B is a rule that assigns to each element x in A exactly one element f(x) in B.

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

• Descriptive Variables: Functions can use variables such as $h$, $a$, $t$ to represent different quantities.

• Output = f(Input): The output is determined by applying the function rule to the input value.

Vertical Line Test

The Vertical Line Test is a graphical method to determine if a relation is a function. If any vertical line crosses the graph of a relation more than once, the relation is not a function.

• Application: Used to verify if a graph represents a function.

One-to-One Functions and the Horizontal Line Test

A function is one-to-one if each output value is paired with exactly one input value. The Horizontal Line Test determines if a function is one-to-one: if any horizontal line crosses the graph more than once, the function is not one-to-one.

• Inverse Functions: Only one-to-one functions have inverses that are also functions.

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Interval Notation: Used to describe domains and ranges, e.g., $(-\infty, \infty)$.

Toolkit Functions

Several basic functions serve as building blocks for more complex functions:

• Constant Function: $f(x) = c$

• Identity Function: $f(x) = x$

• Absolute Value Function: $f(x) = |x|$

• Quadratic Function: $f(x) = x^2$

• Cubic Function: $f(x) = x^3$

• Reciprocal Function: $f(x) = \frac{1}{x}$

• Square Root Function: $f(x) = \sqrt{x}$

Transformations of Functions

Functions can be transformed in various ways to produce new graphs:

• Vertical/Horizontal Shifts: Moving the graph up/down or left/right.

• Reflections: Flipping the graph over the x-axis or y-axis.

• Stretches and Compressions: Changing the steepness or width of the graph.

Function Operations and Composition

• Operations: Functions can be added, subtracted, multiplied, or divided.

• Composition: The composition $f(g(x))$ means applying $g$ first, then $f$ to the result.

Inverse Functions

An inverse function reverses the operation of the original function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Finding Inverses: Swap $x$ and $y$ in the equation and solve for $y$.

Polynomial Functions

General Form and Properties

• General Form: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$

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

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

End Behavior

The end behavior of a polynomial function describes how the function behaves as $x$ approaches $\infty$ or $-\infty$.

• Even Degree: Both ends go in the same direction.

• Odd Degree: Ends go in opposite directions.

Standard vs Vertex Form

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Finding the Vertex: The vertex of a quadratic is at $x = -\frac{b}{2a}$

Quadratic Formula

The quadratic formula solves $ax^2 + bx + c = 0$:

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

Long and Synthetic Division

• Long Division: Used to divide polynomials by other polynomials.

• Synthetic Division: A shortcut method for dividing by linear factors of the form $x - c$.

Rational Functions

A rational function is a function of the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.

• Vertical Asymptotes: Occur where the denominator is zero ($q(x) = 0$).

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

• Removable Discontinuities (Holes): Occur where a factor cancels from numerator and denominator.

Piecewise and Power Functions

• Piecewise Functions: Defined by different expressions for different intervals of the domain.

• Power Functions: Functions of the form $f(x) = x^n$ for some real number $n$.

Summary Table: Key Function Types

Function Type	Equation	Domain	Range
Constant	$f(x) = c$	$(-\infty, \infty)$	$\{c\}$
Identity	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$
Absolute Value	$f(x) = |x|$	$(-\infty, \infty)$	$[0, \infty)$
Quadratic	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Cubic	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$
Reciprocal	$f(x) = \frac{1}{x}$	$(-\infty, 0) \cup (0, \infty)$	$(-\infty, 0) \cup (0, \infty)$
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$