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:

  1. Functions – Function Notation
  2. Functions – Difference Quotient
  3. 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:

Image by Irania Vazquez

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.