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 + 10xNote 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) = −−−−135.
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 PropertiesA. 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) = 3x−4, g(x) = x3 + 4 We first do
(
f g)
(x).(
f g)
(x) = f(g(x)) = f(x3 + 4) Now substitute this in for x in f.= 3(x3+4)− =4 3x3+ − =4 4 3x3 =x So this is half right. Now we do
(
g f)
(x).(
g f)
(x) = g(f(x)) = g(3x−4) = (3x−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)) = f14x+45 = 414x+45−5 = 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) = 3x−5
First, we replace f(x) with y.
y = 3x−5
Now we switch x & y.
x = 3y−5
Now we solve for y.
x3 = y − 5 OR x3 + 5 = y Answer: f −−−−1 (x) = x3 + 5
Verify:
(
f f−1)
(x)= f(f −1 (x)) = f(x3 + 5) = 3( )
x3+5 −5 = 3 3x = x. So this is ok.You verify that
(
f−1f)
(x)= x.3. Now you try one: f(x) = x3 + 1 Answer: f −−−−1 (x) = 3x−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)
