Transcript Inverse Functions and their Representations Lesson 5.2 Definition  A function is a set of ordered pairs with no two first elements alike.   f(x) = {

Inverse Functions and
their Representations
Lesson 5.2
Definition

A function is a set of ordered pairs with no two
first elements alike.


f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }
But ... what if we reverse the order of the pairs?


This is also a function ... it is the inverse function
f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }
Example

Consider an element of an electrical circuit
which increases its resistance as a function of
temperature.
T = Temp
R = Resistance
-20
50
0
150
20
250
40
350
R = f(T)
Example

We could also take the view that we wish to
determine T, temperature as a function of R,
resistance.
R = Resistance
T = Temp
50
-20
150
0
250
20
350
40
T = g(R)
Now we would
say that g(R) and
f(T) are inverse
functions
Terminology


If R = f(T) ... resistance is a function of
temperature,
Then T = f-1(R) ... temperature is the inverse
function of resistance.
is read "f-inverse of R“
is not an exponent
it does not mean reciprocal
 f-1(R)


x 1 
1
x
Does This Have An Inverse?

Given the function at the right
Can it have an inverse?
 Why or Why Not?
NO … when we reverse the ordered
pairs, the result is Not a function
 We would say the function is
not one-to-one



A function is one-to-one
when different inputs always
result in different outputs
c  d  f (c)  f (d )
x
Y
1
5
2
9
4
6
7
5
One-to-One Functions

When different inputs produce the same output


Then an inverse of the function does not exist
When different inputs produce different outputs

Then the function is said to be “one-to-one”
c  d  f (c)  f (d )

Every one-to-one function has an inverse

Contrast
f ( x)  x2 and g ( x)  x3
One-to-One Functions

Examples
f ( x)  x3
f ( x)  x2

Horizontal line test?
Finding the Inverse
Given f ( x)  2 x  7
then y  2 x  7
y 7
solve for x
x
2
y 7
1
f  y 
2
Try
x2
y
x2
Composition of Inverse Functions





f ( x)  x and
Consider
f(3) = 27 and f -1(27) = 3
Thus, f(f -1(27)) = 27
and f -1(f(3)) = 3
3
1
f ( x)  x
3
In general f(f -1(n)) = n and f -1(f(n)) = n
(assuming both f and f -1 are defined for n)
Graphs of Inverses


f ( x)  x3 and
f 1 ( x)  3 x
Again, consider
Set your calculator for the functions shown
Dotted style
Use Standard Zoom
Then use Square Zoom
Graphs of Inverses

Note the two graphs are symmetric
about the line y = x
Investigating Inverse
Functions
f ( x)  2 3 x  4

Consider

Demonstrate that these are inverse functions
What happens with f(g(x))?
Define these
What happens with g(f(x))?
functions on your


x3
g ( x)   4
8
calculator and try
them out
Domain and Range



The domain of f is the range of f -1
The range of f is the domain of f -1
Thus ... we may be required to restrict the
domain of f so that f -1 is a function
Domain and Range

Consider the function h(x) = x2 - 9
Determine the inverse function

Problem => f -1(x) is not a function

Assignment



Lesson 5.2
Page 396
Exercises 1 – 93 EOO