Precalculus 1.5 Inverse Functions:
I. Inverse Functions:
A. Let "f " and "g" be two functions such that
f (g(x)) = x for every "x" in the domain of "g"
and
g( f (x)) = x for every "x" in the domain of "f ".
Under these conditions, the function "g" is the INVERSE of the function "f ":
The function "g" is denoted by f -1 (read f-inverse), so:
f ( f -1(x)) = x
AND
f -1( f (x)) = x
The domain of "f " must be equal to the range of f -1
AND
the range of "f " must be equal to the domain of f -1
Example:
Given f (x) = {(1,2), (4,5), (6,7)}
then f -1 (x) = {(2,1), (5,4), (7,6)}
The graphs of "f " and "f -1 are related to each other by the fact the graph of "f -1" is a reflection of the graph of "f " in the line y = x.
Example:
If the function f has values: {(1,0), (2,3), (4,7)}
then f -1 has values of {(0,1), (3,2), (7,4)}
Example:
Find the inverse of f (x) = 4x + 8
1. let f (x) = y, so by substitution we get
y = 4x + 8
2. Switch the x and y
x = 4y + 8
3. Solve for y
x - 8 = 4y
(1/4) x - 2 = y
4. Now y = f -1 so
f -1 = (1/4) x - 2
Example: Show that f(x) = (x + 8)/3 and g(x) = 3x - 8 are inverse functions of each other. Recall that f(g(x)) = x and g(f(x)) = x therefore:
f (g (x)) = ( (3x-8)+8)/ 3 = (3x)/3 = x so f(g(x)) = x
g ( f (x)) = 3((x + 8)/3) - 8 = x + 8 - 8 = x so g (f (x)) = x
so these two functions are inverse functions of each other.
Example: Find the inverse function of f (x) = x3+5
1. y = x3 + 5
2. x = y3 + 5
3. x - 5 = y3
(x - 5)1/3 = y
so
f -1(x) = (x - 5)1/3
Check answer:
f (f -1(x)) = x
((x - 5)1/3)3 + 5 = x - 5 + 5 = x so it checks
II. One - to - one function:
A function is one-to-one if, for "a" and "b" in the domain
f (a) = f (b) implies that a = b.
A function "f " has an inverse function " f " if and only if " f " is one-to-one.
Use the horizontal line test to check if a function is one-to-one. If the equation can have a horizontal line pass through it only once at any value, then the function is one-to-one.
Example: y2 = x,
doing the vertical line test (to see if it is a function) you see that when x = 1, y = -1 or y = 1 so
this is not a function
but by using the horizontal line test, the line never hits twice so therefore this equation has a one-to-one relationship.
Example: y = x3
this is a function and it has a one-to-one relationship so therefore
this is a one-to-one function.
Testing for One-to-one Functions:
If the function f (x) = x3 + 7
Show that f (a) = f (b)
a3 + 7 = b3 + 7
a3 = b3
a = b so yes it is.
Example:
g (x) = 5x2 + 8
5 (a)2 + 8 = 5 (b)2 +8
5a2 = 5b2
a2 = b2
so you could get:
a = b or
-a = b or
a = -b or
-a = -b
so therefore this function is not one-to-one.
