1. Composition of
Functions
And
Inverse
Functions

2. Evaluating a Composition of Two Functions
When two functions are applied in succession,
the resulting function is called the composite
of the two given functions.
Given f(x) = 8x - 1 and g(x) = 3x + 4, find the following:
b) (g o f)(2)
This tells you to find the
value of g(x) first, then use
this value for the function
f(x). Solve for g(2).
a) (f o g)(2)
(g o f)(x) = g(f(x))
(f o g)(x) = f(g(x))
f(x) = 8x - 1
f(2) = 8(2) - 1
= 15
g(x) = 3x + 4
g(2) = 3(2) + 4
= 10
With the value of g(2) = 10,
you now solve for f(10).
(g o f)(2) = g(f(2))
= g(15)
(f o g)(2) = f(g(2))
= f(10)
g(2) = 10
g(x) = 3x + 4
g(15) = 3(15) + 4
= 49
f(x) = 8x - 1
f(10) = 8(10) - 1
= 79
(f o g)(2) = 79
(g o f)(2) = 49

3. Evaluating a Composition of a Function With Itself
Given h(x) = 4x + 3, find the following:
a) (h o h)(-3)
b) (h o h)(x)
(h o h)(x) = h(h(x))
= h(4(x) + 3)
= h(4x + 3)
h(4x + 3) = 4(4x + 3) + 3
= 16x
= 16x + 15
(h o h)(-3) = h(h(-3))
= h(4(-3) + 3)
= h(-9)
h(-9) = 4(-9) + 3
= -33

4. Inverse Functions
The inverse of a function is a function which “undoes” what the
original function did. It’s graph is the reflection of the original
graph across the line y = x. The graph can be found by
interchanging the coordinates of the ordered pairs in the original function.
The inverse of a function is written as f -1(x) and read as
“the inverse of f at x” or “ f inverse of x.”
When x and y are interchanged in the equation
of a function:
The coordinates of the points that satisfy the
equation are interchanged.
The graph of the function is reflected in the line y = x.
To determine the inverse of a function:
Interchange x and y in the equation of the function.
Solve the resulting equation for y.

5. Graphing the Inverse Function
Note: If the ordered pair (3, 6) is on the graph of
the function f(x), then the ordered pair (6, 3)
will be on the graph of the inverse function, f -1(x).
Example : Find the inverse of the function f(x) = 4x - 7.
y = 4x - 7
Interchange the
x and y values.
x = 4y - 7
x + 7 = 4y
x + 7 = y 4
f-1(x)
(-3, 1)
(-7, 0)
y = x
f(x)
(1, -3)
(0, -7)

6. Verify that the functions f(x) = 4x - 7 and
Verifying an Inverse
If two functions f(x) and g(x) are inverses of
each other, then f(g(x)) must equal x AND
g(f(x)) must equal x.
Verify that the functions f(x) = 4x - 7 and
are inverses.
f(g(x)) must be equal to x.
g(f(x)) must also be equal to x.
f(x) = 4x - 7
g(4x - 7)
= (x + 7) - 7
= x
= x
Since f(g(x)) and g(f(x)) are both equal to x, then f(x) and g(x)
are inverses of each other.

7. y = x Graphing a Function and Its Inverse Graph f(x) = x2 + 1
(-2, 5)
(2, 5)
The graphs are
symmetrical about
the line y = x.
(1, 2)
For the function:
(-1, 2)
(5, 2)
Domain:
Range:
(2 , 1)
y > 1
(0, 1)
(1, 0)
For the inverse:
Domain:
Range:
x > 1
(2, - 1)
y = x
(5, -2)
Is the inverse
a function?
Could the domain
of f(x) be restricted
so that the inverse
is a function?

8. To find the inverse of a Function
Use the Horizontal Line Test to see if an inverse function exists. (That is, will the inverse be a function?)
Replace f(x) with y.
Exchange x and y then solve for y.
Replace y with f-1(x).
To check, calculate f(f-1(x)) and f-1(f(x)) to see if they both equal x.
You can also check by graphing. –> Are the graphs symmetric across the line y=x?

9. Replace f(x) with y.
Exchange x and y.
Solve for y.
Replace y with f-1(x).
Check.

10. Check by graphing.

11. The graph of f(x) and the graph of f-1(x) should reflect over the line y=x.
y=f(x)
y=x
y=f-1(x)

12. Check with composition of functions.