5.2 Inverse Functions and their Representations
Inverses of functions represented numerically (by table)
x
f(x)
0
5
5
10
10
15
15
20
Let’s play a perverse game with f (input-output game
backwards).
 the “perverse” function we just demonstrated
 is called the inverse of the function f
 and is denoted by f-1 (which does not mean 1/f)
x
f-1(x)
5
0
10
5
15
10
20
15
 the inverse of a function simply switches inputs and outputs
 (switches the x’s and y’s)
 accordingly, domain f = range f-1 , range f = domain f-1
Let’s do it with this one:
x
0
5
10
15
f(x)
5
10
5
20
 what's the problem here? (STUDENTS RESPOND)
 f(0) = 5 and f(10) = 5 — two inputs with same output
 but if you try to form the inverse, you get one input with
two outputs: not a function!
 mathematically speaking: f is not one-to-one
 when a function is not one-to-one, it has no inverse
 how do you show that a function is not one-to-one, and
therefore has no inverse?
 find two inputs with the same output
5.2-1
x
f(x)
0
5
5
10
10
15
15
20
x
f-1(x)
5
0
10
5
15
10
20
15
Look at the composition f-1 o f :
(f-1 o f)(0) = f-1(f(0)) = f-1(5) = 0
input
output
“round trip” from 0 to 0
(f-1 o f)(5) = f-1(f(5)) = f-1(10) = 5
input
output
“round trip” from 5 to 5
Also, (f o f-1)(5) = f(f-1(5))= 5 and (f o f-1)(15) = 15
Formal definition of the inverse function f-1
if:
f(g(x)) = x for all x in the domain of g, and
g(f(x)) = x for all x in the domain of f
then:
g is the inverse function for f (g = f-1)
5.2-2
Inverses of functions represented as (mapping) diagrams
to get f-1 , switch inputs and outputs = reverse the arrows!!
Note:
f-1(f(1)) = 1
f(f-1(2)) = 2
f-1(f(2)) = 2
f(f-1(4)) = 4
Works just the way a function and its inverse should work!
5.2-3
When is a function defined by a graph one-to-one?
Consider:

y



   






x




f(-2) = 4, f(2) = 4  two different inputs with same output
 not one-to-one  no inverse
If a horizontal line can cut a graph in more than one place, it
is not one-to-one, and has no inverse:
The horizontal line test:
is one-to-one
is not one-to-one
5.2-4
Inverses of functions represented symbolically (formula)
Remember: f-1 simply switches x (input) and y (output) of f
The method for finding inverses
Example: f(x) = 2x + 1
find f-1:
(1)
(2)
(3)
(4)
y = 2x + 1
x = (y - 1)/2
y = (x - 1)/2
f -1(x) = (x - 1)/2
write it temporarily as:
solve for x:
switch x and y:
rewrite in func. notation
f -1(5) = 2
Does this work? Look: f(2) = 5
Inputs and outputs are switched!
Verify what we did by using the definition of f-1 (on board)
Inverses of functions represented graphically
For the above functions:
y
f(x) = 2x + 1



   





x

f-1(x) = (x - 1)/2



The line in the middle is the line y = x.
Notice:
 f and f-1 are reflections of each other across the line y = x.
 this will be true for any function and its inverse
5.2-5