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
