inverse fun

1
Functions

• A function each element of the domain is paired
  with exactly one element of the range.
• Another way of saying it is that there is one and
  only one output (y) with each input (x).

f(x)
y
x
2
Functions

• Domain the inputs, independent variables, most
  often the xs
• Range the outputs, dependent variables,
• most often the ys

3
Function Notation
Input
Name of Function
Output
4
Determine whether each relation is a function.

• 1. (2, 3), (3, 0), (5, 2), (4, 3)
• YES, every domain is different!

5
Determine whether the relation is a function.

• 2. (4, 1), (5, 2), (5, 3), (6, 6), (1, 9)

NO, 5 is paired with 2 numbers!
6
Is this relation a function?(1,3), (2,3), (3,3)

1. Yes
2. No

7
Vertical Line Test (pencil test)

• If any vertical line passes through more than one
  point of the graph, then that relation is not a
  function.
• Are these functions?

FUNCTION!
FUNCTION!
NOPE!
8
Vertical Line Test
FUNCTION!
NO!
NO WAY!
FUNCTION!
9
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

10
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

11
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

12
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

13
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

14
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

15
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

16
Match the Function

• What type of function is depicted?
• A) Linear
• B) Quadratic
• C) Absolute Value
• D) Exponential Growth
• E) Exponential Decay
• F) Sine
• G) Cosine
• H) Tangent

17
functions---taking a set X and mapping into a Set
Y
1
2
2
4
3
6
4
8
10
5
Set X
Set Y
An inverse function would reverse that process
and map from SetY back into Set X
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One-to-One Functions

• A function is a one-to-one function if no outputs
  are shared (each y-value corresponds to only one
  x-value)
• Passes the horizontal line test

19
If we map what we get out of the function back,
we wont always have a function going back.
1
2
2
4
3
6
4
8
5
Since going back, 6 goes back to both 3 and 5,
the mapping going back is NOT a function These
functions are called many-to-one functions
Only functions that pair the y value (value in
the range) with only one x will be functions
going back the other way. These functions are
called one-to-one functions.
20
This would not be a one-to-one function because
to be one-to-one, each y would only be used once
with an x.
1
2
2
4
3
6
4
8
10
5
This is a function IS one-to-one. Each x is
paired with only one y and each y is paired with
only one x
Only one-to-one functions will have inverse
functions, meaning the mapping back to the
original values is also a function.
21
Recall that to determine by the graph if an
equation is a function, we have the vertical line
test.
If a vertical line intersects the graph of an
equation more than one time, the equation graphed
is NOT a function.
This is NOT a function
This is a function
This is a function
22
To be a one-to-one function, each y value could
only be paired with one x. Lets look at a
couple of graphs.
Look at a y value (for example y 3)and see if
there is only one x value on the graph for it.
For any y value, a horizontal line will only
intersection the graph once so will only have one
x value
This is a many-to-one function
This then IS a one-to-one function
23
If a horizontal line intersects the graph of an
equation more than one time, the equation graphed
is NOT a one-to-one function and will NOT have an
inverse function.
This is NOT a one-to-one function
This is NOT a one-to-one function
This is a one-to-one function
24
Using a graphing calculator, graph the pairs of
equations on the same graph. Sketch your results.
Be sure to use the negative sign, not the
subtraction key.
These graphs are said to be inverses of each
other.
What do you notice about the graphs?
25
An inverse relation undoes the relation and
switches the x and y coordinates.

• In other words, if the relation has coordinates
  (a, b), the inverse has coordinates of (b,a)

Function f(x)
Inverse of Function f(x)
X Y
0 3
1 4
-3 0
-5 2
2 5
-8 5
X Y
3 0
4 1
0 -3
2 -5
5 2
5 -8
26
Notice that the x and y values traded places for
the function and its inverse.
These functions are reflections of each other
about the line y x
Lets consider the function
and compute some values and graph them.

This means inverse function
x f (x)
(2,8)
-2 -8-1 -1 0 0 1
1 2 8
(8,2)
x f -1(x)
-8 -2-1 -1 0 0 1
1 8 2
Lets take the values we got out of the function
and put them into the inverse function and plot
them
(-8,-2)
(-2,-8)
Yes, so it will have an inverse function
Is this a one-to-one function?
What will undo a cube?
A cube root
27
Domain of f
Range of f

• The domain of the original function becomes the
  range of the inverse function.
• The range of the original function becomes the
  domain of the inverse function.

28
Steps for Finding the Inverse of a One-to-One
Function
29
Ensure f(x) is one to one first. Domain may
need to be restricted.
Find the inverse of
30
Ensure f(x) is one to one first. Domain may
need to be restricted.
Find the inverse of
The radicand and range must be positive
31
Ensure f(x) is one to one first. Domain may
need to be restricted.
Find the inverse of
32
Ensure f(x) is one to one first. Domain may
need to be restricted.
Find the inverse of
33
Ensure f(x) is one to one first. Domain may
need to be restricted.
Find the inverse of