Functions and Their Properties in College Algebra

Functions

Definition of a Function

A function is a rule that assigns to each element in a set X (called the domain) exactly one element in a set Y (called the codomain). If an element y in Y is assigned to x in X, it is denoted as f(x). The set of all possible outputs f(x) is called the range of the function.

• Domain: The set of all inputs for which the function is defined.

• Range: The set of all outputs of the function.

• Notation: $f: X \rightarrow Y$

Example: Simple Interest Function

Suppose money is invested at 6% simple interest. The interest earned after time t is given by:

$I = 100(0.06)t$

• For each value of t, there is exactly one value of I.

• We often write $I(t) = 100(0.06)t$ to emphasize the dependence of I on t.

Independent and Dependent Variables

In a function $f: X \rightarrow Y$, the variable that takes values in the domain X is called the independent variable. The variable that represents the output is called the dependent variable or output.

• Example: In $I(t) = 100(0.06)t$, t is independent, I is dependent.

• Example: In $y = x + 2$, x is independent, y is dependent.

Function Rule and Uniqueness

A function must assign exactly one output to each input. For example:

• For $y = x + 2$, each x gives one y.

• For $y^2 = 9$, $y = \pm 3$ when $x = 9$, so one input gives two outputs. This is not a function.

Domain of Functions

Definition and Restrictions

The domain of a function $f: X \rightarrow Y$ is the set of all inputs for which the function makes sense. When X and Y are real numbers, the domain often refers to arithmetic restrictions.

• Example: $h(x) = \frac{1}{x-6}$

• The denominator cannot be zero, so $x \neq 6$.

• Domain: $(-\infty, \infty) \setminus \{6\}$

Equality of Functions

Definition

Two functions $f, g: X \rightarrow Y$ are equal ($f = g$) if:

• The domain of $f$ is equal to the domain of $g$.

• For all $x$ in the domain, $f(x) = g(x)$.

Example: Determining Equality

Consider the following functions:

• $f(x) = \frac{x-1}{x-1}$ if $x \neq 1$; $f(x) = 1$ if $x = 1$

• $g(x) = 1$ for all $x$

To determine equality, check both the domain and the output for all $x$.

Finding Domains

Example

• $f(x) = \sqrt{x+2}$

• Domain: $x + 2 \geq 0 \implies x \geq -2$

• So, domain is $[-2, \infty)$

Polynomial Functions

Constant Functions

A constant function is a function of the form $h(x) = c$, where $c$ is a constant. The domain is all real numbers, and every output is the same.

• Example: $h(x) = 2$

• $h(10) = 2$, $h(-387) = 2$, $h(x+3) = 2$

General Polynomial Functions

A polynomial function is of the form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

• $n$ is a nonnegative integer (the degree).

• $a_n$ is the leading coefficient.

• Example: $f(x) = 3x^2 - 8x + 9$ is degree 2, leading coefficient 3.

Linear and Quadratic Functions

• Linear function: Degree 1, e.g., $f(x) = 2x + 3$

• Quadratic function: Degree 2, e.g., $f(x) = x^2 - 4x + 7$

Zero Function

• $f(x) = 0$ for all $x$

• Called the zero function; by convention, it has no degree.

Rational Functions

Definition

A rational function is a quotient of polynomial functions:

$f(x) = \frac{p(x)}{q(x)}$

• Domain: All real numbers except where $q(x) = 0$

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

Case-Defined Functions

Definition

A case-defined function specifies its rule by cases, depending on the input value.

• Example: $F(s) = \begin{cases} 1 & \text{if } -1 < s < 1 \\ 0 & \text{if } 1 < s < 2 \\ s-3 & \text{if } 2 < s < 8 \end{cases}$

• Domain: $-1 < s < 8$

• To find $F(0)$, use the first case: $F(0) = 1$

• To find $F(2)$, use the second case: $F(2) = 0$

• To find $F(7)$, use the third case: $F(7) = 7 - 3 = 4$

Absolute-Value Function

Definition

The absolute-value function is defined as:

$f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

• Domain: All real numbers

Operations on Functions

Combining Functions

Given functions $f$ and $g$, we can define:

• Sum: $(f+g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x)g(x)$

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

The domain of the new function is the intersection of the domains of $f$ and $g$, except for the quotient, where $g(x) \neq 0$.

Composition of Functions

Definition

The composition of $f$ with $g$ is denoted $f \circ g$ and defined by:

$(f \circ g)(x) = f(g(x))$

• The domain of $f \circ g$ is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Example

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

Summary Table: Types of Functions

Type	General Form	Domain	Example
Constant	$h(x) = c$	$(-\infty, \infty)$	$h(x) = 2$
Linear	$f(x) = ax + b$	$(-\infty, \infty)$	$f(x) = 3x + 1$
Quadratic	$f(x) = ax^2 + bx + c$	$(-\infty, \infty)$	$f(x) = x^2 - 4x + 7$
Polynomial	$f(x) = a_n x^n + \cdots + a_0$	$(-\infty, \infty)$	$f(x) = 2x^3 - x + 5$
Rational	$f(x) = \frac{p(x)}{q(x)}$	All $x$ where $q(x) \neq 0$	$f(x) = \frac{2x+3}{x^2-1}$
Absolute Value	$f(x) = |x|$	$(-\infty, \infty)$	$f(x) = |x|$
Case-Defined	Piecewise	Depends on cases	$F(s)$ as above