5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:46

Review of Precalculus 61

Equivalent Statements

Given a functionf:

1. The functionf has an inverse.


2. The functionf is one-to-one.


3. Every horizontal line passes through the graph offno more than once.


4. The functionf is monotonic strictly increasing or decreasing.

To find the inverse of a functionf:

1. Check iff has an inverse, that is, iffis one-to-one or passes the horizontal line test.


2. Replacef(x)byy.


3. Interchange the variablesxandy.


4. Solve fory.


5. Replaceybyf−^1 (x).


6. Indicate the domain off−^1 (x) as the range of f(x).


7. Verifyf−^1 (x) by checking if f(f−^1 (x))= f−^1 (f(x))=x.

Example 1

Given the graph off(x) in Figure 5.3-5, find:

(a) f−^1 (0)
(b) f−^1 (1)
(c) f−^1 (3)

3

f(x)

(^021)
x
y
1
2
3
Figure 5.3-5
Solutions
(a) By inspection,f(3)=0. Thus, f−^1 (0)=3.
(b) Since f(1)=1, thus,f−^1 (1)=1.
(c) Sincef(0)=3, therefore,f−^1 (3)=0.
Example 2
Determine if the given functions have an inverse:
(a) f(x)=x^3 +x− 2
(b) f(x)=x^3 − 2 x+ 1
Solutions
(a) By inspection, the graph off(x)=x^3 +x−2 in Figure 5.3-6 is strictly increasing, which
implies thatf(x) is one-to-one. (You could also use the horizontal line test.) Therefore,
f(x) has an inverse function.