# 1.8 Review and Summary

## Additional Resources

Access the following online video resources for additional instruction and practice with functions.

## Key Equations

Constant function | [latex]f\left(x\right)=c[/latex], where [latex]\text{}c\text{}[/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 Terms

**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

## 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 1.2 Determining Whether a Relation Represents a Function.
- Function notation is a shorthand method for relating the input to the output in the form [latex]\text{}y=f\left(x\right)\text{}[/latex]. See 1.3 Using Function Notation.
- In tabular form, a function can be represented by rows or columns that relate to input and output values. See 1.4 Representing Functions Using Tables.
- 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 1.5 Finding Input and Output Values of a Function.
- To solve for a specific function value, we determine the input values that yield the specific output value. See 1.5 Finding Input and Output Values of a Function.
- An algebraic form of a function can be written from an equation. See 1.5 Finding Input and Output Values of a Function.
- Input and output values of a function can be identified from a table. See 1.5 Finding Input and Output Values of a Function.
- Relating input values to output values on a graph is another way to evaluate a function. See 1.5 Finding Input and Output Values of a Function.
- A function is one-to-one if each output value corresponds to only one input value. See 1.6 Determining Whether a Function is One-to-One.
- A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See 1.6 Determining Whether a Function is One-to-One.
- The graph of a one-to-one function passes the horizontal line test. See 1.6 Determining Whether a Function is One-to-One.

Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions