Summary: Characteristics of Functions and Their Graphs
Key Equations
| Constant function | [latex]f\left(x\right)=c[/latex], where [latex]c[/latex] is a constant |
| Identity function | [latex]f\left(x\right)=x[/latex] |
| Absolute value function | [latex]f\left(x\right)=|x|[/latex] |
| Quadratic function | [latex]f\left(x\right)={x}^{2}[/latex] |
| Cubic function | [latex]f\left(x\right)={x}^{3}[/latex] |
| Reciprocal function | [latex]f\left(x\right)=\frac{1}{x}[/latex] |
| Reciprocal squared function | [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex] |
| Square root function | [latex]f\left(x\right)=\sqrt{x}[/latex] |
| Cube root function | [latex]f\left(x\right)=\sqrt[3]{x}[/latex] |
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.
- Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\left(x\right)[/latex].
- In tabular form, a function can be represented by rows or columns that relate to input and output values.
- 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.
- To solve for a specific function value, we determine the input values that yield the specific output value.
- An algebraic form of a function can be written from an equation.
- Input and output values of a function can be identified from a table.
- Relating input values to output values on a graph is another way to evaluate a function.
- function is one-to-one if each output value corresponds to only one input value.
- A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
- The graph of a one-to-one function passes the horizontal line test.
Glossary
- dependent variable
- an output variable
- domain
- the set of all possible input values for a relation
- function
- a relation in which each input value yields a unique output value
- horizontal line test
- a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once
- independent variable
- an input variable
- input
- each object or value in a domain that relates to another object or value by a relationship known as a function
- one-to-one function
- a function for which each value of the output is associated with a unique input value
- output
- each object or value in the range that is produced when an input value is entered into a function
- range
- the set of output values that result from the input values in a relation
- relation
- a set of ordered pairs
- vertical line test
- a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
