Table of Contents

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 3: Functions by formulas and graphs

**Topic**. This lesson covers Session 3: Functions by formulas and graphs

**Learning Outcomes.**

- Understand and use function notation.
- Identify domain of a function based on the formula or graph, and the range of a function based on the graph.
- Compute the difference quotient of a function.

**WeBWorK**. There are three WeBWorK assignments on today’s material:

- Functions – Function Notation
- Functions – Difference Quotient
- Functions – Piecewise

**Additional Video Resources.**

**Question of the Day:** If we are looking at the graph of a function, how can we determine the domain and range?

## Functions given by formulas and graphs

**Example 1**.

a. Find the domain and range of the piecewise function:

$$f(x)=\left\{\begin{array}{ll}x^2-1, & \text { for }-1 \leq x \leq 1 \\ x-1, & \text { for } 1<x \leq 5\end{array}\right.$$

b. Find the domain and range of the piecewise function shown in the graph:

*VIDEO: Examples 1&2 – Piecewise Functions*

*Video by Irania Vazquez*

## The difference quotient

**Definition**. The **different quotient** of a function $f(x)$ refers to either one of the following two expressions:

$$\frac{f(x+h)-f(x)}{h}, \quad \frac{f(x)-f(a)}{x-a}$$

If we are given a function $f(x)$, we can use the definition of the function to substitute in the numerator of the difference quotient and then simplify the result.

*Why does this matter? You’ll find that the difference quotient has a very important meaning in Calculus*.

**Example 2.** Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^2-2x+5$.

**Example 3.** Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=\frac{1}{x}$.

*VIDEO: Examples 2 & 3 – Difference Quotients*

*Video by Irania Vazquez*

#### Exit Question

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=2x^2-7$.

#### Answer

$4x+2h$

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.