3.1 Functions and function notation  (Page 9/21)

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in [link] .

Toolkit Functions
Name	Function	Graph
Constant	$f\left(x\right)=c,$ where $c$ is a constant
Identity	$f\left(x\right)=x$
Absolute value	$f\left(x\right)=\left|x\right|$
Quadratic	$f\left(x\right)=x^2$
Cubic	$f\left(x\right)=x^3$
Reciprocal	$f\left(x\right)=\frac{1}{x}$
Reciprocal squared	$f\left(x\right)=\frac{1}{x^2}$
Square root	$f\left(x\right)=\sqrt{x}$
Cube root	$f\left(x\right)=\sqrt[3]{x}$

Key equations

Key concepts

• 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 [link] and [link] .
• Function notation is a shorthand method for relating the input to the output in the form $y=f\left(x\right).$ See [link] and [link] .
• In tabular form, a function can be represented by rows or columns that relate to input and output values. See [link] .
• 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 [link] and [link] .
• To solve for a specific function value, we determine the input values that yield the specific output value. See [link] .
• An algebraic form of a function can be written from an equation. See [link] and [link] .
• Input and output values of a function can be identified from a table. See [link] .
• Relating input values to output values on a graph is another way to evaluate a function. See [link] .
• A function is one-to-one if each output value corresponds to only one input value. See [link] .
• A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See [link] .
• The graph of a one-to-one function passes the horizontal line test. See [link] .

Section exercises

Verbal