Transcript 2.1 Functions and their Graphs

Functions and
their Graphs
Relations
• A relation is a mapping of input values with output
values.
• The set of x-values (input values) is called the
domain.
• The set of y-values (output values) is called the
range.
• A relation is a function provided there is exactly
one output for each input. Each element of the
domain is paired with only one element of the
range
• It is NOT a function if at least one input has more
than one output
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
-2
4
1
4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1
Identify the Domain and Range. Then tell if
the relation is a function.
Input
Output
-3
3
1
1
3
-2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped
onto exactly one output
Vertical Line Test
• You can use the vertical line test to
visually determine if a relation is a
function.
• Slide any vertical line across the graph to
see if any two points lie on the same
vertical line.
• If there are not two points on the same
vertical line then the relation is a function.
• If there are two points on the same
vertical line then the relation is NOT a
function
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(4,4)
(2,2)
(2,-2)
Function?
No, Two points are on
The same vertical line.
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line
Graphing and Evaluating Functions
• Many functions can be represented by an equation
in 2 variables: y = 2x - 7 or! f(x) = 2x - 7
• An ordered pair is a solution if the equation is true
when the values of x & y are substituted into the
equation.
• Ex: (2,-3) is a solution of y = f(x) = 2x-7 because:
• -3 = 2(2) – 7
• -3 = 4 – 7
• -3 = -3
• In an equation, the input variable is called the
independent variable.
• The output variable is called the dependent variable
and depends on the value of the input variable.
• In f(x) = 2x-7 ….. x is the independent variable and y
is the dependent variable
• The graph of an equation in 2 variables is the
collection of all points (x,y) whose coordinates are
solutions of the equation.
Graphing an equation in 2 variables
1. Construct a table of
values
2. Graph enough solutions
to recognize a pattern
3. Connect the points with
a line or curve
Graph: f(x)=y = x + 1
Step
3:
Step2:
Step 1
Table of values
Practice
Create a table with 5 different values.
Graph the lines on the coordinate
plane.
1.f(x) = 2x+ 3
2.h(x) = - 3x+1
3.g(x)= 5 – x
2.R(x)= x - 4
More Practice!
1.
Given f(x) = 9x - 1, find f(0).
2.
If h(x) = -3, find x in h(x) = 7x + 4
3.
Mary is machine saleswoman who earns a base salary of $3,000
plus a commission $200 for every machine she sells. Write a
functions (equation) that shows the total income Mary earns if
she sells x machines in one month. How much money will Mary
make in April if she sells 11 machines?
4.
Paul opens a savings account with $350 dollars. He saves $150
per month. Assume that he does not withdraw money or make
any additional deposits.
a). Write a linear model that represents the total amount of
money Paul has in his account after m months.
b). After how many months will Paul have more than $2,000?
1. f(0))=9(0) – 1 = - 1
2. If h(x) = -3, in h(x) = 7x + 4, then -3 = 7x + 4
Solve for x and x = -1
3. Mary’s salary: y = 3,000 + 200x (x the number of
machines she sells, y her monthly salary)
If Mary sells x = 11 machines, then
y = 3,000 + 200(11) = $5,200
4. a) y = 350 + 150m (m = month, y = money in
savings account)
b) 2000 = 350 + 150m and solve for m;
m = 11 months