Section 2.1
Basics of Functions and Their Graphs
Definition: The domain is the set of _____-values and the range is the set of _____-values.
Definition of a relation: A relation is correspondence from the domain to the range. In other words, it shows
how x and y values relate to each other.
We can be presented a relation in the form of a set of ordered pairs. (𝑥, 𝑦)
{(1,2), (3,4), (5,6), (7,6)}
DOMAIN:
RANGE:
Definition of a Function: A function is a special type of relation such that each element in the domain
corresponds to exactly one element in the range.
Example:
Example 1: Tell whether the following relations can also be considered to be functions.
a) Domain
Range
1
4
2
5
3
6
Function?
c)
Domain
b) Domain
1
2
3
Function?
1
2
1
3
d) {(1,2), (3,4), (5,6), (7,6)}
Function?
4
5
Function?
Range
Range
e) {(2,3), (2,4), (5,6), (7,6)}
Function?
6
How can we tell if a graph represents a function?
Vertical Line Test:
If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.
Example 2: Are the following graphs functions?
a)
b)
If a graph is the graph of a function, then any vertical line will only cross the graph at most once. If even one
vertical line intersects the graph two or more times, the graph is NOT the graph of a function.
Example 3: Use the vertical line test to identify graphs in which y is a function of x.
Function Notation: We replace “y” with the new notation “𝑓(𝑥)” said as “f of x.”
𝒇(𝒙) = 𝒚
“f” is the name of the function;
“x” is the input of the function;
“y” is the output of the function
independent variable
2
𝑓 (𝑥 ) = 𝑥 − 2𝑥 + 1;
4
2
𝑔 (𝑥 ) = 𝑥 + 𝑥 + 3 ;
dependent variable
ℎ(𝑥 ) =
8𝑥 3 +1
𝑥3
;
𝑚(𝑟) = √4 − 𝑟 − 5
Example 4: Evaluate each function at the given value of the independent variable and simplify.
𝑓 (𝑥 + 3)
𝑔(−2)
𝑔(−𝑥 )
ℎ(−2𝑎)
𝑚(−5)
𝑚(4 − 5𝑥 )
Determining Function Values from a graph
𝑔 (𝑥 ) = 𝑦
𝑠𝑎𝑚𝑒 𝑎𝑠
(𝑥, 𝑦)
Use the graph of 𝑔 to answer the following questions.
𝑔(−6)
𝑔(−4)
𝑔(0)
𝑔(7)
𝑔(8)
For what value of 𝑥 is….
𝑔 (𝑥 ) = 9
𝑔(𝑥 ) = −6
𝑔 (𝑥 ) = 0
The zeros of a function 𝑔 are the x-values for which 𝑔(𝑥 ) = 0. These are also called the x-intercepts.
Determine Domain and Range from a graph
Example 5: For each of the following graphs, state the domain and range in interval notation.
DOMAIN:
RANGE:
𝑓(0) =
For what values of 𝑥 is 𝑓(𝑥 ) = 3 ?
What are the zeros of this function f ?
DOMAIN:
RANGE:
𝑓 (−4) =
For what values of 𝑥 is 𝑓(𝑥 ) = 1 ?
DOMAIN:
RANGE:
𝑓 (−2) =
For what values of 𝑥 is 𝑓(𝑥 ) = 2 ?
DOMAIN:
RANGE:
𝑓(2) =
For what values of 𝑥 is 𝑓 (𝑥 ) = −2 ?
Example 6: The human immunodeficiency virus, or HIV, infects and kills helper T cells. Because T
cells stimulate the immune system to produce antibodies, their destruction disables the body's
defenses against other pathogens. By counting the number of T cells that remain active in the body,
the progression of HIV can be monitored. The fewer helper T cells, the more advanced the disease.
The figure below shows a graph that is used to monitor the average progression of the disease. The
average number of T cells, f(x), is a function of time after infection, x.
Developing functions that model given conditions
The cost function:
𝐶 (𝑥 ) = 𝑓𝑖𝑥𝑒𝑑 𝑐𝑜𝑠𝑡 + 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑐𝑜𝑠𝑡.
Fixed costs, such as rent or other overhead, generally remain level, they do not rise or fall directly with the
number of products manufactured.
Variable costs will correlate with the number of products manufactured, including raw materials, packaging,
and labor directly involved in a company's manufacturing process.
Example 7: A company that manufactures bicycles has a fixed cost of $50,000. It costs $500 to produce each
bicycle.
Write the total cost, C, as a function of the number of bicycles produced, x. Then find and interpret C(20).