8G Slides - Chapter 10
Functions
10.1.1 – Functions
• Key Skill: We will be able to examine equations and tables and identify
Key Vocabulary
What are Functions?
• Functions show relationships between inputs and outputs.
• The formula C∘ = (5/9)(F∘ - 32) shows the relationship between degrees
Functions as Machines
• Think of a machine that doubles whatever number you input.
– Input is 2, output is 4
– Input is 100, output is 200 – Input is -40, output is -80
Function Example
• Degrees Celsius versus Fahrenheit:
• Note that for each input, there is ONLY ONE possible output.
Fahrenheit 0 32 70 100 125 212
Contra-Example
• Imagine a machine that takes the square root of each input
• Is this a function?
Contra-Example
• Imagine a machine that takes the square root of each input
• Is this a function?
• No!, for each input there are TWO
outputs. If ‘x’ is 4, y is BOTH 2 and -2
Functions and Tables
• Consider the table below. Is it a function?
x 0 1 2 3 4 5
Functions and Tables
• Consider the table below. Is it a function?
• Yes! Each input has ONLY ONE output. The equation is y = 4x + 7
x 0 1 2 3 4 5
Another Example
• Is this relationship a function?
x 0 1 2 3 4 5
Another Example
• Is this relationship a function?
• Yes! There is still ONLY ONE output for each input. The equation is y = 7
x 0 1 2 3 4 5
Another Example
• Is this relationship a function?
x 0 1 2 3 4 5
Another Example
• Is this relationship a function?
• No! There are TWO outputs for each input. The equation is y = 2x ⎜ ⎜
x 0 1 2 3 4 5
Classwork
10.1.2 – Functions and Graphs
Functions
• Remember that for a relationships to be a function, there can be ONLY ONE output for each input.
Graphical Example
Graphical Example
• Does this graph represent a function?
Graph Example
Graph Example
• Is this relationship a function?
Vertical Line Test
• If you can move a Vertical Line across the graph and NEVER have the line
Example
Example
Classwork
Function Notation
Function Notation
• We are used to seeing linear
relationships shown in the form y = 2x + 1
• Function notation is similar, but
replaces the ‘y’ with f(x) as in f(x) = 2x + 1
Function Notation
Function Notation
• If f(x) = 2x + 1 then what is f(3)? • To find f(3), we simply insert the
number 3 into the equation as follows: f(x) = 2(3) + 1 and solve.
Non-Linear Example
Non-Linear Example
• Functions can also be non-linear. • If f(x) = 2x2 what is f(5)?
Examples
• If f(x) = -3x – 6 find f(4)
• If g(x) = (2/3)x + 11 find g(12)
Examples
• If f(x) = -3x – 6 find f(4)
f(9) = -3(4) – 6 or -18
• If g(x) = (2/3)x + 11 find g(12)
g(12) = (2/3)(12) + 11 or 19
• If h(x) = -2x3 – 4 find g(2)
Challenge
Challenge
• If f(x) = 2x + 4 and g(x) = -3x - 8 then… what is f(g(5)) ?
Classwork/Homework
Domain and Range
Domain
• What are the possible values of ‘x’ in the following function?
Domain
• What are the possible values of ‘x’ in the following function?
• ‘x’ can only be a non-negative number • We say the Domain of ‘x’ is x ≥ 0
Key Vocabulary
Examples
• Find the Domains for the following functions:
f
(
x
)
= -
2
x
+
1
Examples
• Find the Domains for the following functions:
f
(
x
)
= -
2
x
+
1
g
(
x
)
=
3 /
x
x = all real numbers
Range
• What are the possible outputs in the following function:
Range
• What are the possible outputs in the following function:
f(x) = x
⎜ ⎜
• The outputs are all non-negative.
We say the Range includes all
Key Vocabulary
Examples
• Find the ranges of the following functions:
f
(
x
)
= -
2
x
+
1
Examples
• Find the ranges of the following functions:
f
(
x
)
= -
2
x
+
1
f(x) = all real numbersg(x) ≥ 0
From Graphs
From Graphs
• Find domain and range:
From Ordered Pairs
• Find the domain and range of the following relationship:
From Ordered Pairs
• Find the domain and range of the following relationship:
• { (1,3), (2,6), (3,9), (4,12) } • Domain: {1, 2, 3, 4}
