Functions and Models: Foundations for Calculus

Functions and Models

Objectives

• Determine whether a correspondence is a function.

• Find function values.

• Graph functions and determine whether a graph represents a function.

• Graph functions that are piecewise-defined.

Definition of a Function

Basic Concept

A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

• Domain: The set of all possible input values (independent variable, usually x).

• Range: The set of all possible output values (dependent variable, usually f(x)).

Determining Whether a Correspondence is a Function

Examples

• Example 1a: Number of iPhones Sold Yearly

  Domain (Year)	Range (Millions Sold)
  2019	187.2
  2020	196.9
  2021	242
  2022	250

  • This is a function because each year corresponds to only one sales figure.

• Example 1b: Squaring

  Domain	Range
  3	9
  4	16
  5	25
  -5	25

  • This is a function because each input has only one output, even if different inputs share the same output.

• Example 1c: Basketball Teams

  Domain (City)	Range (Team)
  New York	Knicks
  Los Angeles	Lakers, Clippers
  Atlanta	Hawks

  • This is not a function because Los Angeles corresponds to two teams (Lakers and Clippers).

Evaluating Functions

Finding Function Values

• Given a function rule, substitute the input value to find the output.

• Example: For $f(x) = x^2$, find $f(-3)$, $f(1)$, $f(k)$, $f(\sqrt{k})$, and $f(x+h)$.

  • $f(-3) = (-3)^2 = 9$

  • $f(1) = 1^2 = 1$

  • $f(k) = k^2$

  • $f(\sqrt{k}) = (\sqrt{k})^2 = k$

  • $f(x+h) = (x+h)^2 = x^2 + 2xh + h^2$

• Example: For $f(x) = 3x + 5$, find $f(4)$, $f(-5)$, $f(0)$, and $f(x+h)$.

  • $f(4) = 3(4) + 5 = 17$

  • $f(-5) = 3(-5) + 5 = -10$

  • $f(0) = 3(0) + 5 = 5$

  • $f(x+h) = 3(x+h) + 5 = 3x + 3h + 5$

Difference Quotient

Definition and Examples

The difference quotient is a fundamental concept for understanding rates of change and derivatives:

$\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x - x^2$, find $f(4)$, $f(x+h)$, and $\frac{f(x+h) - f(x)}{h}$.

  • $f(4) = 4 - 4^2 = -12$

  • $f(x+h) = x + h - (x+h)^2 = x + h - (x^2 + 2xh + h^2) = x - x^2 - 2xh - h^2 + h$

  • $\frac{f(x+h) - f(x)}{h} = \frac{[x - x^2 - 2xh - h^2 + h] - [x - x^2]}{h} = \frac{-2xh - h^2 + h}{h} = -2x - h + 1$ (for $h \neq 0$)

• Example: For $f(x) = 2x - x^2$, $\frac{f(x+h) - f(x)}{h} = 2 - 2x - h$

Graphs of Functions

Definition and Construction

The graph of a function $f$ is the set of all points $(x, f(x))$ in the coordinate plane. If the function is given by an equation, its graph is the graph of $y = f(x)$.

• To graph a function:

  1. Choose values for $x$ (inputs).

  2. Compute the corresponding $y = f(x)$ (outputs).

  3. Plot the points $(x, y)$ on the coordinate plane.

  4. Connect the points smoothly if appropriate.

• Example: Graph $f(x) = x^2 - 1$.

  x	f(x)	(x, f(x))
  -2	3	(-2, 3)
  -1	0	(-1, 0)
  0	-1	(0, -1)
  1	0	(1, 0)
  2	3	(2, 3)

The Vertical-Line Test

Determining if a Graph Represents a Function

• A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• This test ensures that each input $x$ has only one output $y$.

• Example: Some graphs (like $y = x^3$) pass the test and are functions; others (like a circle) do not.

Piecewise-Defined Functions

Definition and Graphing

A piecewise-defined function is a function defined by different expressions for different intervals of the domain.

• Example: $g(x) = \begin{cases} 3, & x = 1 \\ -x + 2, & x \neq 1 \end{cases}$

  • For $x = 1$, $g(1) = 3$.

  • For $x \neq 1$, $g(x) = -x + 2$.

  • Graph the line $y = -x + 2$ with an open circle at $x = 1$, and plot the point $(1, 3)$.

  x	g(x)	(x, g(x))
  -3	5	(-3, 5)
  0	2	(0, 2)
  1	3	(1, 3)
  2	0	(2, 0)
  3	-1	(3, -1)

• Example: $f(x) = \begin{cases} 1, & x = -2 \\ 2 - x, & x \neq -2 \end{cases}$

  • For $x = -2$, $f(-2) = 1$.

  • For $x \neq -2$, $f(x) = 2 - x$.

  • Graph the line $y = 2 - x$ with an open circle at $x = -2$, and plot the point $(-2, 1)$.

  x	f(x)	(x, f(x))
  -3	5	(-3, 5)
  -2	1	(-2, 1)
  0	2	(0, 2)
  1	1	(1, 1)
  2	0	(2, 0)

Summary Table: Function vs. Not a Function

Situation	Is it a Function?	Reason
Each input has one output	Yes	Meets definition
One input has two outputs	No	Fails definition
Different inputs share the same output	Yes	Allowed