Functions and Their Properties in College Algebra

Functions

Definition of a Function

A function from a set X to a set Y, denoted as f: X → Y, is a rule that assigns to each element x in X exactly one element f(x) in Y. The set X is called the domain of the function, and the set of all possible outputs f(x) is called the range of the function.

• Domain: The set of all input values x for which f(x) is defined.

• Range: The set of all output values f(x) for some x in the domain.

Example: Simple Interest Function

If money is invested at 6% simple interest, the interest earned after time t years is given by:

$I = 100(0.06)t$

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

• Independent variable: t (time)

• Dependent variable: I (interest earned)

Independent and Dependent Variables

In a function f: X → Y:

• Independent variable: The variable that takes values in the domain (X).

• Dependent variable: The variable whose value depends on the independent variable, often called the output.

Example: Linear Function

The equation $y = x + 2$ defines y as a function of x. For each value of x, there is exactly one value of y.

• If $x = 1$, then $y = 1 + 2 = 3$.

• Independent variable: x

• Dependent variable: y

Non-Function Example

Not all equations define functions. For example, $y^2 = 9$ gives $y = ext{±}3$ for $x = 9$, assigning two outputs to one input, which violates the definition of a function.

Domain of Functions

Definition of Domain

The domain of a function f: X → Y is the set of all x for which f(x) makes sense as an element of Y. For real-valued functions, the domain often refers to values of x that do not cause mathematical issues (such as division by zero).

Example: Rational Function Domain

Consider $h(x) = \frac{1}{x - 6}$.

• The function is undefined when $x = 6$ (division by zero).

• Domain: All real numbers except 6, i.e., $(-\infty, \infty) - \{6\}$.

Equality of Functions

Definition

Two functions f and g from X to Y are equal (denoted $f = g$) if:

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

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

Example: Piecewise Functions

Consider the following piecewise definitions:

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

• $f(x) = 1$ if $x = 1$

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

• $g(x) = 1$ if $x = 1$

To determine if $f = g$, check both the domain and the output for each input.

Finding Domains

Example: Square Root Function

For $f(x) = \sqrt{x - 2}$, the domain is all $x$ such that $x - 2 \geq 0$, i.e., $x \geq 2$.

Example: Linear Function

For $g(x) = x + 5$, any real number can be used for $x$, so the domain is $(-\infty, \infty)$.

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 value.

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

All function values are the same, so $h(x)$ is a constant function.

General Polynomial Functions

A polynomial function in $x$ 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 of the polynomial).

• $a_n$ is the leading coefficient.

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

Linear and Quadratic Functions

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

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

Zero Function

The zero function $f(x) = 0$ is a polynomial with no degree assigned by convention. Its domain is all real numbers.

Rational Functions

Definition

A rational function is a quotient of two polynomial functions:

$R(x) = \frac{P(x)}{Q(x)}$

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

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

Case-Defined Functions

Definition

A case-defined function (or piecewise function) specifies different rules for different intervals of the independent variable.

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

• For $s = 0$, $F(0) = 1$; for $s = 2$, $F(2) = 0$; for $s = 7$, $F(7) = 4$.

Absolute-Value Function

Definition

The absolute-value function is defined as $f(x) = |x|$:

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

• Domain: All real numbers.

Operations on Functions

Combining Functions

Given functions $f$ and $g$, new functions can be formed by:

• 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 combined function is the intersection of the domains of $f$ and $g$, except for the quotient, which further excludes values where $g(x) = 0$.

Composition of Functions

Definition

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

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

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

Example: Composite Function

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	$h(x) = c$	All real numbers	$h(x) = 2$
Linear	$f(x) = ax + b$	All real numbers	$f(x) = 3x + 1$
Quadratic	$f(x) = ax^2 + bx + c$	All real numbers	$f(x) = x^2 - 4x + 7$
Polynomial	$f(x) = a_n x^n + ... + a_0$	All real numbers	$f(x) = 2x^3 - x + 5$
Rational	$f(x) = \frac{P(x)}{Q(x)}$	All real numbers except $Q(x) = 0$	$f(x) = \frac{2x+3}{x^2-1}$
Absolute Value	$f(x) = |x|$	All real numbers	$f(x) = |x|$
Piecewise	Defined by cases	Depends on cases	See above

Additional info: Some explanations and examples have been expanded for clarity and completeness.