1.2 – FUNCTIONS AND THEIR PROPERTIES (DAY 1)
I.
Definitions
A.
B.
C.
D.
Domain – set of first elements in a relation; input values; x-values.
Range – set of second elements in a relation; output values; y-values.
Relation – set of ordered pairs
Function – a relation where for each domain value, there is a unique y-value (domain
does not repeat.
Example 1: Determine if the following are functions. (Exer. 1-4)
a) 𝑥 2 + 𝑦 = 4
b) 𝑥 2 + 𝑦 2 = 4
c) 𝑥 2 + 𝑦 3 = 8
Conclusion: If the degree of 𝑦 is odd, the expression will be a function. If the degree of 𝑦 is
even, it will not be a function because you end up with ±.
You can also look at the graph to see if it’s a function using the vertical line test. If the graph
crosses any vertical line more than once, it is not a function.
Example 2: Use the vertical line test to determine whether the curve is the graph of a
function (Exer. 5-8)
a)
b)
c)
The domain of a function can be restricted for 2 reasons that you need to be aware of in this course:
1.
2.
NO negatives … no negative numbers inside a square root.
**
… no zeros in the denominator of a fraction.
NO zeros
Example 3: Determine the domain and write answer in interval notation (Exer. 9-16)
a) 𝑓 𝑥 = 3𝑥 2 − 𝑥 + 5
The domain of a function can be restricted for 2 reasons that you need to be aware of in this course:
1.
2.
NO negatives … no negative numbers inside a square root.
**
… no zeros in the denominator of a fraction.
NO zeros
Example 3: Determine the domain and write answer in interval notation (Exer. 9-16)
b) 𝑓 𝑥 =
3
𝑥+4
The domain of a function can be restricted for 2 reasons that you need to be aware of in this course:
1.
2.
NO negatives … no negative numbers inside a square root.
**
… no zeros in the denominator of a fraction.
NO zeros
Example 3: Determine the domain and write answer in interval notation (Exer. 9-16)
c) 𝑓 𝑥 = 𝑥 − 1
The domain of a function can be restricted for 2 reasons that you need to be aware of in this course:
1.
2.
NO negatives … no negative numbers inside a square root.
**
… no zeros in the denominator of a fraction.
NO zeros
Example 3: Determine the domain and write answer in interval notation (Exer. 9-16)
d) 𝑓 𝑥 =
𝑥
𝑥−5
The domain of a function can be restricted for 2 reasons that you need to be aware of in this course:
1.
2.
NO negatives … no negative numbers inside a square root.
**
… no zeros in the denominator of a fraction.
NO zeros
Example 3: Determine the domain and write answer in interval notation (Exer. 9-16)
e) 𝑓 𝑥 = 𝑥 2 − 9
To find the range, look at the graph of the function. Look at the y-values.
Example 4: Determine the range of the function. Write answer in interval notation. (Exer.
17-20)
a) f(x) =
2
𝑥
To find the range, look at the graph of the function. Look at the y-values.
Example 4: Determine the range of the function. Write answer in interval notation. (Exer.
17-20)
b) 𝑓 𝑥 = 5𝑥 − 3
Continuity
A function is continuous at a point if the graph does not come apart at that point.
non-removable discontinuity
Increasing/Decreasing Functions
Read the graph from LEFT to RIGHT. Make sure you are looking at x-values!!!
Example 5: Identify where the graph is increasing, decreasing, or constant. (Exer. 25-34)
a)
b)
Identifying Local Extrema
Maximum: occurs when function changes from increasing to decreasing.
Minimum: occurs when function changes from decreasing to increasing.
Example 6: State whether each labeled point is a local min, local max, or neither.
a)
b)
Homework: Pg. 102 problems 1-34