Chapter 2
Section 2.3
Functions
Function Definition, Domain and Range and Notation
x 32
y 6 2 25
y 6
12
2
x
32
Domain of a Function:
(formula) The values for 𝑥 you can "plug into" the
formula.
0 ≤ 25 − 𝑥 − 3 2
25 ≤ 𝑥 − 3 2
−5 ≤ 𝑥 − 3 ≤ 5
−2 ≤ 𝑥 ≤ 8
(graph) Where the graph is above, below or on the xaxis. Looking at the graph it is the interval −2,8 .
12
10
4
25
10
8
8
6
6
4
4
2
2
2
4
6
8
Not a Function!
4
2
2
4
6
8
Is a Function!
A vertical line can only intersect
the graph once.
Any vertical line only intersects
the graph once.
When 𝑥 = 6 the value for 𝑦 can
be either 2 or 10.
When 𝑥 = 6 then 𝑦 must equal
10.
Range of a Function:
(graph) Where the graph is to the left, right or on the
y-axis. Looking at the graph it is the interval 6,11 .
Notation:
To say "y is a function of x" , y is the dependent
variable, x is the independent variable, we write:
𝑦=𝑓 𝑥 .
𝑓 𝑥 = 6 + 25 − 𝑥 − 3
2
𝑡ℎ𝑒𝑛 𝑓 6 = 10
Example 2.3.1:
If 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 5 find each of the following: 𝑓 −2 , 𝑓
1
3
, 𝑓 3𝑥 , and 𝑓 𝑥 + 1 .
𝑓 −2 = −2 2 − 3 −2 + 5 = 4 + 6 + 5 = 15
2
1
1
1
1
37
𝑓
=
−3
+5= −1+5=
3
3
3
9
9
𝑓 3𝑥 = 3𝑥 2 − 3 3𝑥 + 5 = 9𝑥 2 − 9𝑥 + 5
𝑓 𝑥 + 1 = 𝑥 + 1 2 − 3 𝑥 + 1 + 5 = 𝑥 2 + 2𝑥 + 1 − 3𝑥 − 3 + 5 = 𝑥 2 − 𝑥 + 3
Example 2.3.2:
Express y as a function of x (i.e. 𝑦 = 𝑓 𝑥 )where y is the bottom half of the parabola 𝑥 = 3 − 𝑦 − 2 2 . Find the domain, range
and compute the value of 𝑓 −6 .
4
𝑥 =3− 𝑦−2 2
𝑦−2 2 =3−𝑥
𝑦−2=− 3−𝑥
(bottom half)
𝑦 =2− 3−𝑥
𝑓 𝑥 =2− 3−𝑥
Domain:
0≤3−𝑥
𝑥≤3
Range:
𝑦≤2
3
Interval: −∞, 2
2
Interval: −∞, 3
𝑓 −6 = 2 − 3 − −6
=2− 9
= −1
1
1
1
2
3
Increasing and Decreasing Intervals for a Function
10
Increasing on an Interval:
8
A function is increasing on an interval if as the values for x
increase the values for y also increase.
6
Graphically it means the graph goes "uphill" as you move
from left to right.
4
2
Decreasing on an Interval:
1
1
2
3
4
2
Increasing: 1,3
Decreasing: −∞, 1 ⋃ 3, ∞
5
A function is increasing on an interval if as the values for x
increase the values for y also increase.
Graphically it means the graph goes "downhill" as you move
from left to right.