# Chapter 3: Introduction and Outcomes

**Functions**

A relation associates two elements. For example we might relate time with distance where we would discuss how far something travels in a given amount of time. A function is a special type of relation where each independent value is matched with only one dependent value. In our example, time would be the independent value and distance would be the dependent value because you can determine how far you might travel given an amount of time. The independent value is our input (domain) and the dependent value is our output (range).

In this chapter, functions will be examined along with their properties. A library of functions will be introduced and will be used to create piecewise functions and to transform functions. The graphs of functions will be discussed along with the associated properties. Functions will be combined using algebraic operations and compositions. The concept of one-to-one functions will be used to determine if a function has an inverse and then where applicable, the inverse function will be found.

**Chapter Sections:**

- 3.1: Functions and Function Notation
- 3.2: Domain and Range
- 3.3: Rates of Change and Behavior of Graphs
- 3.4 Composition of Functions
- 3.5: Transformation of Functions
- 3.6: Absolute Value Functions
- 3.7: Inverse Functions

**1.** **Functions: **Successful College Algebra students demonstrate a deep understanding of functions whether they are described verbally, numerically, graphically, or algebraically (both explicitly and implicitly). Students should be proficient working with the following families of functions: linear, quadratic, higher-order polynomial, rational, exponential, logarithmic, radical, and piecewise-defined functions (including absolute value).

**1a. **Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.). In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).

**1b. **Convert between different representations of a function.

**1c. **Perform operations with functions including addition, subtraction, multiplication, division, composition, and inversion; connect properties of constituent functions to properties of the resultant function; and resolve a function into a sum, difference, product, quotient, and/or composite of functions.

**4. Modeling with Functions:** Successful College Algebra students should have experience in using and creating mathematics which model a wide range of phenomena.

**4a.** Create linear models from data and interpret slope as a rate of change.

**Chapter Section Objectives and related OTM standards:**

3.1: Functions and Function Notation

- Determine whether a relation represents a function. (1a)
- Find the value of a function. (1c)
- Determine whether a function is one-to-one. (1c)
- Use the vertical line test to identify functions. (1a)
- Graph the functions listed in the library of functions (1a)

3.2: Domain and Range

- Find the domain of a function defined by an equation. (1a)
- Graph piecewise-defined functions. (1a, 1b)

3.3: Rates of Change and Behavior of Graphs

- Find the average rate of change of a function. (4b)
- Use a graph to determine where a function is increasing, decreasing, or constant. (1a)
- Use a graph to locate local maxima and local minima. (1a)
- Use a graph to locate the absolute maximum and absolute minimum. (1a)

3.4 Composition of Functions

- Combine functions using algebraic operations. (1c)
- Create a new function by composition of functions. (1c)
- Evaluate composite functions. (1c)
- Find the domain of a composite function. (1c)
- Decompose a composite function into its component functions. (1c)

3.5: Transformation of Functions

- Graph functions using vertical and horizontal shifts. (1b)
- Graph functions using reflections about the x-axis and the y-axis. (1b)
- Determine whether a function is even, odd, or neither from its graph. (1b)
- Graph functions using compressions and stretches. (1b)
- Combine transformations. (1b)

3.6: Absolute Value Functions

- Graph an absolute value function. (1a)
- Solve an absolute value equation. (N/A)

3.7: Inverse Functions

- Verify inverse functions. (1c)
- Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one. (1c)
- Find or evaluate the inverse of a function. (1c)
- Use the graph of a one-to-one function to graph its inverse function on the same axes. (1c)