Essential Concepts

Functions and Function Notation

A function is a special type of relation where each input (domain value) corresponds to exactly one output (range value).

The notation [latex]f(x)[/latex] is read as “f of x” and represents the output value when x is the input. For example, if [latex]f(x) = x^2 + 1[/latex], then [latex]f(3) = 3^2 + 1 = 10[/latex].

Determining if a Relation is a Function

• From ordered pairs or tables: Check that no x-value repeats with different y-values
• From a graph: Use the vertical line test—if any vertical line intersects the graph more than once, it’s not a function

Evaluating Functions

• Algebraically: Substitute the input value for the variable and simplify
• From a table: Find the input value and read the corresponding output
• From a graph: Locate the x-value and find the corresponding y-coordinate

A function is one-to-one if each output corresponds to exactly one input. Use the horizontal line test: if any horizontal line intersects the graph more than once, the function is not one-to-one.

Domain and Range

The domain is the set of all possible input values (x-values) for a function

The range is the set of all possible output values (y-values) for a function

Finding Domain from an Equation

Exclude values that would cause:

1. Division by zero: Set denominators ≠ 0
2. Even roots of negatives: Set expressions under even roots ≥ 0
3. For functions with both: Combine all restrictions

Finding Domain and Range from a Graph

• Domain: Read the x-values from left to right where the graph exists
• Range: Read the y-values from bottom to top where the graph exists

Interval Notation Tips

• Use [ ] for included endpoints (≤ or ≥)
• Use ( ) for excluded endpoints (< or >)
• Always write from smaller to larger value
• Use [latex]\cup[/latex] to combine separate intervals

A piecewise function uses different formulas for different parts of the domain. To evaluate:

1. Determine which interval your input falls in
2. Use the formula for that interval
3. Substitute and evaluate

To graph a piecewise function, graph each piece on its specified interval, using open circles for excluded endpoints and closed circles for included endpoints.

Rates of Change and Behaviors of Graphs

The average rate of change measures how quickly a function’s output changes relative to its input over an interval:

[latex]\text{Average Rate of Change} = \dfrac{f(x_2) - f(x_1)}{x_2 - x_1} = \dfrac{\Delta y}{\Delta x}[/latex]

The units are “output units per input unit” (like miles per hour or dollars per year).

• If a function is increasing, the values go up as you move left to right (positive rate of change)
• If a function is decreasing, the values go down as you move left to right (negative rate of change)
• If a function is constant, then the values stay the same (zero rate of change)

Local and Absolute Extrema

• Local maximum: A point higher than nearby points (like a hilltop)
• Local minimum: A point lower than nearby points (like a valley)
• Absolute maximum: The highest point on the entire graph
• Absolute minimum: The lowest point on the entire graph

Note: When stating where extrema occur, give the x-value. When stating the extrema value, give the y-value (or [latex]f(x)[/latex] value).

Key Equations

Average Rate of Change: [latex]\dfrac{f(x_2) - f(x_1)}{x_2 - x_1}[/latex]

Glossary

absolute maximum

the largest output value over the entire domain of a function

absolute minimum

the smallest output value over the entire domain of a function

average rate of change

the change in output divided by the change in input over a specified interval

decreasing function

a function where output values decrease as input values increase

dependent variable

the output variable in a function, often represented by y or [latex]f(x)[/latex]

domain

the set of all possible input values for a function

function

a relation where each input corresponds to exactly one output

horizontal line test

a test to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it’s not one-to-one

increasing function

a function where output values increase as input values increase

independent variable

the input variable in a function, often represented by x

input

a value from the domain that is entered into a function

interval notation

a way of writing sets of numbers using brackets and parentheses to show included and excluded endpoints

local maximum

a point where the function value is greater than all nearby points

local minimum

a point where the function value is less than all nearby points

one-to-one function

a function where each output value corresponds to exactly one input value

output

the value produced when an input is entered into a function

piecewise function

a function defined by different formulas over different parts of the domain

range

the set of all possible output values for a function

rate of change

a measure of how one quantity changes relative to another

relation

a relationship between two sets of numbers

vertical line test

a test to determine if a graph represents a function; if any vertical line intersects the graph more than once, it’s not a function