# Functions and Inverse Functions ( ) ( ) ( ) { ```Functions and Inverse Functions
Definition of a function is a set of order pairs for which there is exactly one value of y for each
value of x, given that the order pair is ( x, y ) . For example, a set of points such as {(1,3) , (2,4 )}
would be considered a function, that is for each value of x would be assigned a unique value of y.
This next set would not be a function {(1,3) , (1,4)} , notice that one value of x has two different y
values. A better way to see this is plotting the points a making a graph; the Vertical Line Test can
help identify which graphs are functions.
Below are sets of points to illustrate the definition of a function. Suppose you have sets of
ordered pairs such as:
Set A
A = {(− 3,−2 ) , (- 1,1) , (1,1) , (2,1) , (3,4 )}
Draw a straight vertical line through the points. The
graph of set A does not hit twice on any vertical line
drawn through the graph.
This is a Function.
B = {(1,1) , (1,3) , (1,-2) , (1,4 ) , (3,4 )}
Draw a straight vertical line at (1,1) . The graph of set B is
hitting more than one point at x=1.
This is Not a Function.
C = {(1,1) , (- 1,2 ) , (- 2,-2) , (1,1) , (3,4 )}
Although (1,1) repeats in set C, it’s still a function.
There are no repeating y values for one single x.
This is a Function.
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Set B
Set C
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The following are examples to practice and helpful to understand what defines a
function. The Vertical Line test is used to check if these are functions.
Not a Function
Function
Not a Function
Function
Function
Not a Function
Function
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Inverse of a Function
If f (x ) and g ( x ) are functions such that f [g ( x )] = x and g [ f ( x )] = x , then g ( x ) is the inverse
of f ( x ) and g ( x ) is notated as f −1 ( x ) .
Example 1:
Suppose f ( x ) = x − 2 and g ( x ) = x 2 + 2
Check to see if g ( x ) is an inverse function of f (x ) .
2nd: find g [ f ( x )] = x
1st: find f [g ( x )] = x
g (x ) = x 2 + 2
f (x ) = x − 2
Substitute x 2 + 2 in for g ( x )
f [g ( x )] = f x 2 + 2
[
[
] (x
f x +2 =
f [g ( x )] = x
2
]
2
)
+2 −2 = x
Substitute
x − 2 in for f (x )
[ x − 2]
g[ x − 2 ] = ( x − 2 ) + 2 = x − 2 + 2
g [ f ( x )] = g
2
2
g [ f ( x )] = x
f [g ( x )] = x and g [ f ( x )] = x , by definition g ( x ) is the inverse function of f ( x ) , so g ( x ) = f −1 ( x ) .
In order to find an inverse of a function; graphically substitute the x values for the y values.
Simply change all the x variables into y’s and all the y variables into x’s.
The following is an example: the points flip on the y=x, in other words, the points reflect over the
line y=x. The inverses are displayed with squares in Graph A.
A = {(− 3,−2) , (− 1,1) , (1,1) , (2,1) , (3,4)}
Graph A
Set A
x
-3
-1
1
2
3
y
-2
1
1
1
4
A − 1 = {(− 2,−3) , (1,−1) , (1,1) , (1,2 ) , (4,3)}
Set A −1
1x
4
-2
1
1
2y
3
-3
-1
1
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The following are two examples to find inverses of a function.
x−2
4
Substitute all x variables for y’s and y
variables for x’s.
y−2
x−2
h( x ) = y =
⇒x=
4
4
Find the inverse of h( x ) =
Find the inverse of f ( x ) = 2 x 3 + 2
Substitute all x variables for y’s and y variables
for x’s.
f (x ) = y = 2 x 3 + 2 ⇒ x = 2 y 3 + 2
Then Solve for y.
(4)x = ⎛⎜ y − 2 ⎞⎟ (4)
⎝ 4 ⎠
4x = y − 2
4x + 2 = y − 2 + 2
4x + 2 = y
Then solve for y.
x = 2 y3 + 2
x − 2 = 2 y3 + 2 − 2
x − 2 = 2y3
x − 2 2y3
=
2
2
x−2
= y3
2
Take the cube root of both sides of equation.
x−2 3 3
3
= y
2
3
x−2
=y
2
From this example, the inverse of the
function is h −1 ( x ) = 4 x + 2 .
The graph below illustrates.
⇒
The inverse of the function is f −1 ( x ) = 3
x−2
2
The graph below illustrates.
f (x )
y=x
y=x
h −1 ( x )
f
−1
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h( x )
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