1-2 Functions
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Write the equation of the line in slope
intercept as described
1) through : (-1,5) , parallel to y=3x-5
2) through :(0,3) , perpendicular to 2x-y=4
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Write the equation of the line in slope
intercept as described
1) through : (-1,5) , parallel to y=3x-5
Answer: 𝑦 = 3𝑥 + 8
2) through :(0,3) , perpendicular to 2x-y=4
Answer: 𝑦 =
1
− 𝑥
2
+3
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Student will be able:
Determine whether a relation between two
variables represents a function .
Use function notation and evaluate functions.
Find the domains of functions.
Use functions to model and solve real-life
problems.
Evaluate difference quotients.
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What is a function?
A function f from a set A to a set B is a
relation that assigns to each element x in the
set A exactly one element y in the set B.
The set A is the domain (or set of inputs) of
the function f, and the set B contains the
range (or set of outputs).
What is a relation?
Any set of ordered pairs
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A function consists of three things:
I) A set called the domain
ii) A set called the range
iii) A rule which associates each element of
the domain with a unique element of
the range.
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Each element in A must be matched with an
element in B.
2. Some elements in B may not be matched
with any element in A.
3. Two or more elements in A may be
matched with the same element in B.
4. An element in A (the domain) cannot be
matched with two different elements in B.
Function
Not a function
function
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Let A={2,3,4,5} and B={-3,-2,-1,0,1}.Which
of the following sets of ordered pairs
represent functions from set A to set B.
a.){(2,-2),(3,0),(4,1),(5,-1)}
b.){(4,-3),(2,0),(5,-2),(3,1)(2,-1)}
c.){(2,-1),(3,-1),(4,-1),(5,-1)}
d.){(3,2),(5,0),(2,-3)}
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Lets say we have this equation
𝑦 = 𝑥2
In this equation , x, is the independent
variable and y is the dependent variable.
the domain of the function is the set of all
values taken by x.
The range of the function is the set of all
values taken on by y.
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To determine whether an equation represents y
as a function of x. we have the following
examples
Example#2: 2x+3y=6
Solution:
First we solvefor y
3𝑦 = −2𝑥 + 6 we move x to the other side
−2𝑥+6
2𝑥
𝑦=
= − + 2 we divide by 3
3
3
Next we prove the equation is a function
2 2
2
Lets say x=2, 𝑦 = −
+ 2 = so y=2/3
3
3
Since we have only one output or y value we can
say the equation is a function
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Example #3: −𝑥 + 𝑦 2 = 1
Solution:
Solve for y
𝑦 2 = 𝑥 + 1 we move the x
𝑦 2 = ∓ 𝑥 + 1 square root
𝑦 =∓ 𝑥+1
Lets say we let x=3
𝑦 = ∓ 3 + 1 = ∓ 4 = ∓2
So y is not a function of x since y can be 2 or
-2
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Example #4: determine if the following
equations are functions:
A.) 3𝑥 2 + 𝑦 2 = 20
B.)
𝑥
−
3
𝑦
2
+ =6
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A function is a rule that takes an input, does
something to it,
and gives a unique corresponding output.
There is a special notation (called ‘function
notation’) that is used to represent this situation:
if the function name is f , and the input name is
x ,
then the unique corresponding output is called
f(x) .
The notation ‘ f(x) ’ is read aloud as: ‘ f of x ’.
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Example #5:
Given f(x) = x2 + 2x – 1, find f(2),f(-3).
f(2) = (2)2 +2(2) – 1
=4+4–1
=7
f(–3) = (–3)2 +2(–3) – 1
=9–6–1
=2
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Evaluating functions practice 1-6
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What is a piecewise function?
Answer: A function defined by two or more
equations over a specified domain.
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Example #6
Evaluate the function for f(0) and f(2).
2𝑥 2 − 1, 𝑥 < 1
𝑓 𝑥 =
𝑥 + 4, 𝑥 ≥ 1
If we want to evaluate f(0), we must, since 0 < 1,
use the first half of the function. Then f(0) =
2(0)2 – 1 = 0 – 1 = –1.
If we want to evaluate f(2), then we must, since 2
> 1, use the second half of the function. Then
f(2) = (2) + 4 = 6
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Example #7
Evaluate the following function for f(4),f(-1)
and f(0)
𝑥 − 3, 𝑥 < 0
𝑓 𝑥 =
𝑥 + 4, 𝑥 ≥ 0
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The height in meters of a projectile at t
seconds can be found by the function ℎ 𝑡 =
− 4.9𝑡 2 + 60𝑡 + 1.2. Find the height of the
projectile 4 seconds after it is launched…
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One of the basic definitions in calculus.
For a function f, the formula
, ℎ ≠ 0.
This formula computes the slope of the
secant line through two points on the graph
of f. These are the points with x-coordinates
x and x + h. The difference quotient is used
in the definition the derivative.
𝑓 𝑥+ℎ −𝑓 𝑥
ℎ
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From Worksheet
Problems 1-12 only
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Today we learn about functions and how to
evaluate them. We also learn about piecewise
functions and the difference of quotients.
Tomorrow we are going to learn about 1-3
graphs of the functions.