Functions and Their Graphs: Library and Piecewise-Defined Functions

Functions and Their Graphs

Library of Basic Functions

The library of functions consists of several fundamental functions that serve as building blocks for more complex functions. Understanding their properties and graphs is essential in precalculus.

The Constant Function

• Definition: $f(x) = c$, where $c$ is a constant.

• Domain: $(-\infty, \infty)$

• Range: $\{c\}$

• x-intercept: None (unless $c = 0$)

• y-intercept: $(0, c)$

• Constant: Everywhere

• Increasing/Decreasing: Neither

• Even/Odd: Even

The Identity Function

• Definition: $f(x) = x$

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, \infty)$

• x-intercept: $(0, 0)$

• y-intercept: $(0, 0)$

• Constant: Nowhere

• Increasing: Everywhere

• Decreasing: Nowhere

• Even/Odd: Odd

The Square Function

• Definition: $f(x) = x^2$

• Domain: $(-\infty, \infty)$

• Range: $[0, \infty)$

• x-intercept: $(0, 0)$

• y-intercept: $(0, 0)$

• Constant: At $x = 0$

• Increasing: $x > 0$

• Decreasing: $x < 0$

• Even/Odd: Even

The Cube Function

• Definition: $f(x) = x^3$

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, \infty)$

• x-intercept: $(0, 0)$

• y-intercept: $(0, 0)$

• Constant: At $x = 0$

• Increasing: Everywhere

• Decreasing: Nowhere

• Even/Odd: Odd

The Square Root Function

• Definition: $f(x) = \sqrt{x}$

• Domain: $[0, \infty)$

• Range: $[0, \infty)$

• x-intercept: $(0, 0)$

• y-intercept: $(0, 0)$

• Constant: At $x = 0$

• Increasing: $x > 0$

• Decreasing: Nowhere

• Even/Odd: Neither

The Cube Root Function

• Definition: $f(x) = \sqrt[3]{x}$

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, \infty)$

• x-intercept: $(0, 0)$

• y-intercept: $(0, 0)$

• Constant: At $x = 0$

• Increasing: Everywhere

• Decreasing: Nowhere

• Even/Odd: Odd

The Absolute Value Function

• Definition: $f(x) = |x|$

• Domain: $(-\infty, \infty)$

• Range: $[0, \infty)$

• x-intercept: $(0, 0)$

• y-intercept: $(0, 0)$

• Constant: At $x = 0$

• Increasing: $x > 0$

• Decreasing: $x < 0$

• Even/Odd: Even

The Reciprocal Function

• Definition: $f(x) = \frac{1}{x}$

• Domain: $(-\infty, 0) \cup (0, \infty)$

• Range: $(-\infty, 0) \cup (0, \infty)$

• x-intercept: None

• y-intercept: None

• Constant: Nowhere

• Increasing: $x < 0$

• Decreasing: $x > 0$

• Even/Odd: Odd

Piecewise-Defined Functions

A piecewise-defined function is a function defined by different expressions on different intervals of its domain. These functions are useful for modeling situations where a rule changes based on the input value.

• General Form: $f(x) = \begin{cases} \text{expression 1} & \text{if } x \text{ in interval 1} \\ \text{expression 2} & \text{if } x \text{ in interval 2} \\ \cdots \end{cases}$

• Example: The absolute value function can be written as a piecewise-defined function: $f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

Example: Evaluating a Piecewise Function

Given $f(x) = \begin{cases} x^2 & \text{if } x < 2 \\ 6 & \text{if } x = 2 \\ 10-x & \text{if } 2 < x \leq 6 \end{cases}$, evaluate $f(-3), f(0), f(2), f(3), f(6), f(7)$.

• $f(-3) = (-3)^2 = 9$

• $f(0) = 0^2 = 0$

• $f(2) = 6$

• $f(3) = 10 - 3 = 7$

• $f(6) = 10 - 6 = 4$

• $f(7)$ is undefined (since $x > 6$ is not in the domain)

Example: Application of Piecewise Functions

Suppose a doctor's fee is based on the length of time:

• Up to 6 minutes costs $50

• Over 6 and up to 15 minutes costs $80

• Over 15 minutes costs $80 plus $5 per minute above 15 minutes

This can be written as:

$C(t) = \begin{cases} 50 & 0 < t \leq 6 \\ 80 & 6 < t \leq 15 \\ 80 + 5(t-15) & t > 15 \end{cases}$

Analyzing Piecewise-Defined Functions

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

• Range: The set of all possible output values.

• Intercepts: Points where the graph crosses the axes.

• Continuity: Whether the graph has any breaks or jumps.

• Increasing/Decreasing: Intervals where the function rises or falls.

• Even/Odd/Neither: Symmetry properties of the function.

Example: Graphing and Analyzing a Piecewise Function

Given $f(x) = \begin{cases} -2x+1 & -3 \leq x < 1 \\ 2 & x = 1 \\ x^2 & x > 1 \end{cases}$:

• Domain: $[-3, \infty)$

• Intercepts: Solve $f(x) = 0$ for each piece.

• Continuity: Check for jumps at $x = 1$.

• Range: Use the graph to determine all possible $f(x)$ values.

Absolute Value as a Piecewise Function

The absolute value function is a classic example of a piecewise-defined function:

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

Practice: Evaluating and Writing Piecewise Functions

• Given $f(x) = |3x-2|$, write as a piecewise function:

• Given $g(x) = 3|x-2| - 5x$, write as a piecewise function.

• Given $h(x) = 2|1-x| - 1$, write as a piecewise function.

Sample Problems

Function	Intervals	Evaluate
$f(x) = \begin{cases} x^3 & -2 \leq x < 1 \\ 3x+2 & 1 \leq x \leq 4 \end{cases}$	$-2 \leq x < 1$, $1 \leq x \leq 4$	$f(-1), f(0), f(1), f(3)$
$f(x) = \begin{cases} -3x & x < -1 \\ 0 & x = -1 \\ 2x^2+1 & x > -1 \end{cases}$	$x < -1$, $x = -1$, $x > -1$	$f(-2), f(-1), f(0)$
$f(x) = \begin{cases} \frac{1}{x} & x < 0 \\ \sqrt[3]{x} & x \geq 0 \end{cases}$	$x < 0$, $x \geq 0$	Evaluate at various $x$
$f(x) = \begin{cases} 2x+5 & -3 \leq x < 0 \\ -3 & x = 0 \\ -5x & x > 0 \end{cases}$	See intervals	Evaluate at various $x$

Additional info: For each piecewise function, always specify the domain for each piece, check for continuity at the endpoints, and graph each segment according to its formula and interval.