*Inverse Relations
*Objectives: Students will be able to…
*Determine whether a function has an inverse
*Write an inverse function
*Verify 2 functions are inverses of each other
3
x f(x)
33333 9999
999
y f-
1(x)
9999999 3333
333
x2 √ x
* The inverse of a given function will “undo” what the original function did.
* For example, let’s take a look at the square function: f(x) = x2
*Inverse Relations:
*The inverse of f is f -1 (read “f inverse”)
*If both the original relation and the inverse relation are both functions, they are inverse functions!
*The domain of the original relation is the range of the inverse.
Graphically, the x and y values of a function and its inverse are switched.
If the function y = g(x) contains the points
then its inverse, y = g-
1(x), contains the points
x 0 1 2 3 4
y 1 2 4 8 16
x 1 2 4 8 16
y 0 1 2 3 4
Where is there a line of reflection?
The graph of a
function and its
inverse are mirror images
about the line
y = x
y = f(x)
y = f-
1(x)
y = x
*Find the inverse relation:
x -3 -1 5 7 10
y -5 -3 0 4 7
x
y
Writing the equation of an inverse function:
Example 1: y = 6x - 12
Step 1: Switch x and y: x = 6y - 12
Step 2: Solve for y: x 6y 12x 12 6yx 126
y
1
6x 2 y
*Find an equation for the inverse relation:
1. 2. 712 xy7
5
7
3 xy
*Verifying 2 functions are inverses
* If f and g are inverse functions, their composition would simply give x back.
xxgfgf xxfgfg
*Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are
inverses.*Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.
*Because f(g(x))=x and g(f(x))=x, they are inverses.
f(g(x))= -3(-1/3x+2)+6
= x-6+6
= x
g(f(x))= -1/3(-3x+6)+2
= x-2+2
= x
2;2,2 2 xxgxxxf
xxxgf 22
22
xxxfg 2222 2
Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.
*Verify that the functions f and g are inverses of each other.
*Remember…
The graph of a function needs to pass the vertical line test in order to be a function.
*Horizontal Line Test
*Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test.
*If the original function passes the horizontal line test, then its inverse will pass the vertical line test and therefore is a function.
*If the original function does not pass the horizontal line test, then its inverse is not a function (may need to restrict domain so that it has an inverse).
*One to One Functions
*All functions have only one y value for any given x value (this means they pass the vertical line test)
*One-to-one functions also have only one x-value for any given y-value.
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 a one-to-one function
This is NOT a one-to-one function
This is NOT a one-to-one function
*Ex: Graph the function f(x)=x2 and determine
whether its inverse is a function.
Graph does not pass the horizontal line test, therefore the
inverse is not a function.
*But what if I restricted the domain of x2? Then it would
have an inverse. Graph y = x2 for x > 0. Find the inverse function.
*Ex: Graph f(x)=2x2-4. Determine whether f -1(x) is a function.
*How could I restrict the domain to make it have an inverse function?
f -1(x) is not a function.
x > 0
Example 2:
Given the function : y = 2x2 -4, x > 0 find the inverse
Step 1: Switch x and y:x = 2y2 -4
Step 2: Solve for y:
yx
yx
yx
yx
2
4
2
4
24
42
2
2
2
Only need the positive root for inverse!
*Ex: Write the inverse of g(x)=2x3
Inverse is a function!
y=2x3
x=2y3
3
2y
x
yx
3
2
3
2
xy
2
43 xy
*Graph the functions to determine whether their
inverses will also be functions