Jay Abramson; Sharon North

Key Concepts

3.1 Functions and Function Notation

A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See Example 1 and Example 2.
Function notation is a shorthand method for relating the input to the output in the form $y = f (x) .$ See Example 3 and Example 4.
In tabular form, a function can be represented by rows or columns that relate to input and output values. See Example 5.
To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See Example 6 and Example 7.
To solve for a specific function value, we determine the input values that yield the specific output value. See Example 8.
An algebraic form of a function can be written from an equation. See Example 9 and Example 10.
Input and output values of a function can be identified from a table. See Example 11.
Relating input values to output values on a graph is another way to evaluate a function. See Example 12.
A function is one-to-one if each output value corresponds to only one input value. See Example 13.
A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See Example 14.
The graph of a one-to-one function passes the horizontal line test. See Example 15.

3.2 Domain and Range

The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
The domain of a function can be determined by listing the input values of a set of ordered pairs. See Example 1.
The domain of a function can also be determined by identifying the input values of a function written as an equation. See Example 2, Example 3, and Example 4.
Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See Example 5.
For many functions, the domain and range can be determined from a graph. See Example 6 and Example 7.
An understanding of toolkit functions can be used to find the domain and range of related functions. See Example 8, Example 9, and Example 10.
A piecewise function is described by more than one formula. See Example 11 and Example 12.
A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See Example 13.

3.3 Rates of Change and Behavior of Graphs

A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See Example 1.
Identifying points that mark the interval on a graph can be used to find the average rate of change. See Example 2.
Comparing pairs of input and output values in a table can also be used to find the average rate of change. See Example 3.
An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See Example 4 and Example 5.
The average rate of change can sometimes be determined as an expression. See Example 6.
A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See Example 7.
A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
Minima and maxima are also called extrema.
We can find local extrema from a graph. See Example 8 and Example 9.
The highest and lowest points on a graph indicate the maxima and minima. See Example 10.

3.4 Composition of Functions

We can perform algebraic operations on functions. See Example 1.
When functions are composed, the output of the first (inner) function becomes the input of the second (outer) function.
The function produced by composing two functions is a composite function. See Example 2 and Example 3.
The order of function composition must be considered when interpreting the meaning of composite functions. See Example 4.
A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
A composite function can be evaluated from a table. See Example 5.
A composite function can be evaluated from a graph. See Example 6.
A composite function can be evaluated from a formula. See Example 7.
The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See Example 8 and Example 9.
Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
Functions can often be decomposed in more than one way. See Example 10.

3.5 Transformation of Functions

A function can be shifted vertically by adding a constant to the output. See Example 1 and Example 2.
A function can be shifted horizontally by adding a constant to the input. See Example 3, Example 4, and Example 5.
Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See Example 6.
Vertical and horizontal shifts are often combined. See Example 7 and Example 8.
A vertical reflection reflects a graph about the $x -$ axis. A graph can be reflected vertically by multiplying the output by –1.
A horizontal reflection reflects a graph about the $y -$ axis. A graph can be reflected horizontally by multiplying the input by –1.
A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See Example 9.
A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See Example 10.
A function presented as an equation can be reflected by applying transformations one at a time. See Example 11.
Even functions are symmetric about the $y -$ axis, whereas odd functions are symmetric about the origin.
Even functions satisfy the condition $f (x) = f (- x) .$
Odd functions satisfy the condition $f (x) = - f (- x) .$
A function can be odd, even, or neither. See Example 12.
A function can be compressed or stretched vertically by multiplying the output by a constant. See Example 13, Example 14, and Example 15.
A function can be compressed or stretched horizontally by multiplying the input by a constant. See Example 16, Example 17, and Example 18.
The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See Example 19 and Example 20.

3.6 Absolute Value Functions

Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See Example 1.
The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See Example 2.
In an absolute value equation, an unknown variable is the input of an absolute value function.
If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See Example 3.

3.7 Inverse Functions

If $g (x)$ is the inverse of $f (x),$ then $g (f (x)) = f (g (x)) = x .$ See Example 1, Example 2, and Example 3.
Only some of the toolkit functions have an inverse. See Example 4.
For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
For a tabular function, exchange the input and output rows to obtain the inverse. See Example 5.
The inverse of a function can be determined at specific points on its graph. See Example 6.
To find the inverse of a formula, solve the equation $y = f (x)$ for $x$ as a function of $y .$ Then exchange the labels $x$ and $y .$ See Example 7, Example 8, and Example 9.
The graph of an inverse function is the reflection of the graph of the original function across the line $y = x .$ See Example 10.