**Section 8.4 - Composite and Inverse Functions**

I. Composition of Functions

A. If f and g are functions, then the composite function of f and g (written f

^{ g) is: }

(f

g)(x) = f(g(x))The domain of f

g is the set of all x in the domain of g such that g(x) is in the domain of f.B. With composition, we are, in effect, substituting a number into g(x), finding out what y is, and then substituting that answer into f(x).

C. Examples - Let f(x) = 9 − 2x, g(x) = −5x + 2. find the following.

1. (f

^{ g)(x) }

First, we use the definition of composition to get:

(f

g)(x) = f(g(x))Now we will substitute into this equation what g(x) is equal to:

(f

g)(x) = f(−5x + 2)Next, we substitute −5x + 2 in for x in f(x), EVEN THOUGH x IS REPEATED!

(f

g)(x) = 9 − 2(−5x + 2) Simplifying, we get:**Answer: (f **

**g)(x) = 5 + 10x**2. Now you try one: (g

^{ f)(x) }

**Answer: (g **

**−−−−**

^{ f)(x) = }**43 + 10x**

Note that composition, in general, is not commutative.

3. (f

^{ g)(3) }

Again, we start by using the definition of composition to get:

(f

g)(3) = f(g(3))Substituting 3 for x in g(x), we get:

(f

g)(3) = f(−5(3) + 2) = f(−13)We now substitute −13 in for x in f(x) to get:

(f

g)(3) = 9 − 2(−13) Simplifying, we get:**Answer: (f **

**g)(3) = 35**4. Now you try one: (g

^{ f)(3) }

**Answer: (g **

^{ f)(3) = }^{−−−−}

^{13 }5.

a. (f

^{ g)( −2) }

We start by using the definition of composition:

(f

g)( −2) = f(g(−2))We now have to determine the value of y when x is −2 for the graph of g(x):

(f

g)( −2) = f(2)We next look at the graph of f(x) and determine the value of y when x is 2:

**Answer: (f **

**g)( −2) = 3**b. Now you try one: (g

^{ f)( −4) }

**Answer: (g **

**f)( −4) = 2**II. Inverse Properties

A. **Recall that for a real number A, the additive inverse was that real number B such that **
A + B = 0.

B. For a real number A ≠** 0, the multiplicative inverse is that real number B such that AB = 1. **

**C. For a function f(x), the inverse function is that function g(x) such that **

### (

^{f }^{g}### )

(x) = x and### (

*g*

^{f}### )

(x) = x.D. Verifying that functions are inverses of each other.

1. Do the composition

### (

^{f }^{g}### )

(x). If the answer is x, you are halfway there.2. Now do the composition

### (

^{g }

^{f}### )

(x). If this answer is also x, then f and g are inverse functions of each other. We then would write that g(x) = f^{ -1}

**(x). The "-1" is NOT an**exponent. This notation means that we have the inverse function of f(x). Note that f(x) is also g

^{ -1}(x).

E. Examples - Determine if f(x) and g(x) are inverses of each other.

1. f(x) = ^{3}*x*−4, g(x) = x^{3} + 4
We first do

### (

*f*

^{g}### )

(x).### (

^{f }^{g}### )

(x) = f(g(x)) = f(x^{3}+ 4) Now substitute this in for x in f.

= ^{3}(*x*^{3}+4)− =4 ^{3}*x*^{3}+ − =4 4 ^{3}*x*^{3} =*x* So this is half right. Now we do

### (

*g*

^{f}### )

(x).### (

^{g }

^{f}### )

(x) = g(f(x)) = g(^{3}

*x*−4) = (

^{3}

*x*−4)

^{3}+ 4 = x - 4 + 4 = x

**Answer: f(x) and g(x) are inverses.**

**y = f(x) ** **y = g(x) **

2. Now you try one: f(x) = 5x − 9, g(x) = 5
9
*x*+

**Answer: f(x) and g(x) are not inverses. **

III. Inverse Functions

A. **For a function f(x), the inverse function is that function g(x) such that **

### (

^{f }^{g}### )

(x) = x and### (

^{g }

^{f}### )

^{(x) = x. }

B. Verifying that functions are inverses of each other.

1. Do the composition

### (

^{f }^{g}### )

(x). If the answer is x, you are halfway there.2. Now do the composition

### (

*g*

^{f}### )

(x). If this answer is also x, then f and g are inverse functions of each other. We then would write that g(x) = f^{−}

^{1}(x). The "−1" is NOT an exponent. This notation means that we have the inverse function of f(x). Note that f(x) is also g

^{−}

^{1}(x).

IV. Determining if a Function has an Inverse

A. **A function f(x) is one-to-one if for every y in the range there is only one x in the domain that **
corresponds to it.

B. **Horizontal Line Test: A function f(x) is not one-to-one if any horizontal line intersects the **
graph of f(x) in more than one point.

C. If a function f(x) is one-to-one, then its inverse is also a function. When this occurs, we write the
inverse function as f ^{-1}** (x) (read "f inverse of x"). Note that this is the functional inverse, NOT **
the multiplicative inverse.

D. Examples - Are these functions one-to-one?

1.

**Answer: Not one-to-one. **

2.

**Answer: Yes one-to-one. **

3. Now you try one:

V. Finding the Inverse

A. In general, to find the inverse of a relation, we switch x & y in the ordered pairs. Remember that x is the domain, y is the range.

B. This means, geometrically, that the graph of a relation and its inverse are reflections of each other across the identity function line, f(x) = x.

C. Finding the inverse of a function f(x)

1. Determine if the function is one-to-one.

2. Write y for f(x).

3. Switch x & y.

4. Solve for y.

5. Write f ^{−}^{1} (x) for y.

6. Verify by showing that

## (

^{f}^{}

^{f}^{−}

^{1}

## ) (

^{(}

^{x}^{)}

^{=}

^{f}^{−}

^{1}

^{}

^{f}## )

^{(}

^{x}^{)}

^{=}

^{x}^{. }

7. Remember:

a. Domain of f is the range of f ^{−}^{1}.
b. Range of f is the domain of f ^{−}^{1}.
D. Examples - Find the inverse function.

1. f(x) = 4x − 5

First, we write y for f(x).

y = 4x − 5

Next, switch x & y.

x = 4y − 5

Now we solve for y.

x + 5 = 4y OR *x* *y*

+ = 4 5

**Answer: f **^{−−−−}^{1}** (x) = **
**4**
**x** **5**
**4**
**1** +
Verify:

## (

^{f }

^{f}^{−}

^{1}

## )

^{(}

^{x}^{)}

^{=}

^{ f(f }

^{−}

^{1}

^{ (x)) = f}

^{}

_{}

^{}

^{1}

_{4}

^{x}^{+}

_{4}

^{5}

^{}

_{}

^{}

^{ = 4}

^{}

_{}

^{}

^{1}

_{4}

^{x}^{+}

_{4}

^{5}

^{}

_{}

^{}−5 = x + 5 − 5 = x So this is ok.

You verify that

## (

^{f}^{−}

^{1}

^{}

^{f}## )

^{(}

^{x}^{)}

^{=}

^{x}^{. }

Note that if we graph these, we make a table for the function that is the "easiest", then switch x & y to get the table for the inverse.

**Answer: Yes one-to-one. **

2. f(x) = ^{3}*x*−5

First, we replace f(x) with y.

y = ^{3}*x*−5

Now we switch x & y.

x = ^{3}*y*−5

Now we solve for y.

x^{3} = y − 5 OR x^{3} + 5 = y
**Answer: f **^{−−−−}^{1}** (x) = x**^{3}** + 5 **

Verify:

## (

^{f }

^{f}^{−}

^{1}

## )

^{(}

^{x}^{)}

^{=}

^{ f(f }

^{−}

^{1}(x)) = f(x

^{3}+ 5) =

^{3}

## ( )

^{x}^{3}

^{+}

^{5}

^{−}

^{5}

^{ = }

^{3 3}

*= x. So this is ok.*

^{x}You verify that

## (

^{f}^{−}

^{1}

^{}

^{f}## )

^{(}

^{x}^{)}

^{=}

^{x}^{. }

3. Now you try one: f(x) = x^{3} + 1
**Answer: f **^{−−−−}^{1}** (x) = **^{3}**x**−**1**
VI. Graphing a function and its inverse.

A. Remember that to find the inverse, we switch x & y.

B. So if we make a table for f(x), to get a table for f^{ −1} (x), we just switch x & y on the table.

C. Examples - Graph f(x) and f^{ −}^{1} (x) on the same set of axes.

1. f(x) = 4x − 5, f ^{−}^{1} (x) =1 5
4x +4

Making a table for f(x) will be relatively easy, but f^{−}^{1} (x) doesn't look so nice!

Now graph both of these.

**x ** **f(x) = 4x **−−−−** 5 **

0 −5

1 −1

**x ** **f **^{−−−−}^{1}** (x) =1** **5**
**4x +4**

−5 0

−1 1

Switch x & y on the table
to get the table for f^{−}^{1} (x).

**f(x) **

**f**^{−−−−}^{1 }**(x) **