Summary: Characteristics of Functions and Their Graphs

Key Equations

Constant function f(x)=cf(x)=c, where cc is a constant
Identity function f(x)=xf(x)=x
Absolute value function f(x)=|x|f(x)=|x|
Quadratic function f(x)=x2f(x)=x2
Cubic function f(x)=x3f(x)=x3
Reciprocal function f(x)=1xf(x)=1x
Reciprocal squared function f(x)=1x2f(x)=1x2
Square root function f(x)=√xf(x)=√x
Cube root function f(x)=3√xf(x)=3√x

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 y=f(x)y=f(x).
  • 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