This document contains lecture notes from a Math 1000 class taught by Stuart Jones. It covers several topics related to functions, including definitions of functions, evaluating functions, piecewise functions, and the difference quotient. Key points covered include the definition of a function as a mapping from a domain to a range, evaluating functions at different inputs, how piecewise functions use different formulas in different intervals, and the formula for the difference quotient.

1. Math 1000
Stuart Jones
Section2.1
Functions

2. Math 1000
Stuart Jones
What is a function?
A function takes a domain D as input and outputs a unique
element from R (the range) for each element of D.

3. Math 1000
Stuart Jones
Evaluate f (3), f (−1), and f (0) if f (x) = −2x2 − 4x − 1

4. Math 1000
Stuart Jones
Evaluate f (2), f (0) and f (a) if f (x) =
√
4x + 1

5. Math 1000
Stuart Jones
The net change from a point x=a to a point x=b is the overall
change in the y coordinate. It is represented as f (b) − f (a).

6. Math 1000
Stuart Jones
Find the net change from x = 2 to x = 6 if f (x) = 2x2 −4x +2

7. Math 1000
Stuart Jones
Find the net change from x = 0 to x = 10 if f (x) = 2−x
x−11

8. Math 1000
Stuart Jones
Piecewise functions are segments of different functions bound
together into one function using a set of rules. For example:
f (x) =
x2 − 4 if x ≤ 0
2x + 1 if x > 0
This means that if x is 0 or less than 0, the top function will
apply. However, if x is above 0, then the bottom function
applies.

9. Math 1000
Stuart Jones
Evaluate f (0), f (−1), and f (1)
f (x) =
x3 − 2x if x ≤ 0
3
√
x if x > 0

10. Math 1000
Stuart Jones
Evaluate f (2), f (3), f (−1), and f (1)
f (x) =



2
x if x ≤ −2
x + 4 if − 2 < x < 1
2−x
x+4 if x ≥ 1

11. Math 1000
Stuart Jones
Functions can also be evaluated at variables and other functions
(you saw this earlier with f(a).) We will talk more about this in
a future section, but if we have f (3x), all x terms get
multiplied by 3, or if we have 3f (x), then the entire function
gets multiplied by 3. Let’s look at a couple of examples.

12. Math 1000
Stuart Jones
If f (x) = x−4
x+2, find f (−4x).

13. Math 1000
Stuart Jones
If f (x) =
√
x − 5, find 3f (x) − 2(f (x))2

14. Math 1000
Stuart Jones The last topic we will discuss in this section is one that will be
important for anyone taking calculus later on. (And you still
need to know it for this class anyway.) It is called the difference
quotient.
Difference Quotient
For any function f (x), the difference quotient is defined as:
f (a + h) − f (a)
h
NOTE: Problems involving the difference quotient will almost
ALWAYS involve simplifying! If you still have an h in the
denominator, you aren’t finished.

15. Math 1000
Stuart Jones
Find the difference quotient of f (x) = 3x − 4

16. Math 1000
Stuart Jones
Find the difference quotient of f (x) = 2x2 − 4x

17. Math 1000
Stuart Jones
Find the difference quotient of f (x) = x
x+2

18. Math 1000
Stuart Jones
The Bottom Line
Piecewise functions will return soon - make sure you
understand them and how they work, or you’ll be very
confused later when we have to graph them.
Memorize the difference quotient formula. You’ll be using
this later in this class and when you get to future classes.