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 table 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.
  • A 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.
  • A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.


dependent variable
an output variable
the set of all possible input values for a relation
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
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
each object or value in the range that is produced when an input value is entered into a function
the set of output values that result from the input values in a 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