CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 1. Functions and Graphs

3. Determine the inverse off(x) =x+x 4.
   Answers:


4. To find the inverse,

y= 5 + 2 x

x= 5 + 2 y

x− 5 =y 2

2 (x− 5 ) =y=f−^1 (x)

Verification:
2 ( 5 +x 2 − 5 )= 2 (x 2 )=x

5 +^2 (x 2 −^5 )= 5 +x− 5 =x
They are truly inverses of each other.

2. Even thoughf(x) =^37 x−21 andg(x) =^73 x+21 have some inverted pieces, they are not inverses of each other.
   In order to show this, you must show that the composition does not simplify tox.^37 (^73 x+ 21 )− 21 =x+ 9 − 21 =
   x− 126 =x


3. To find the inverse, switch x and y.

f(x) =y=x+x 4

x=y+y 4

x(y+ 4 ) =y

xy+ 4 x=y

xy−y=− 4 x

y(x− 1 ) =− 4 x

f−^1 (x) =y=−x^4 −x 1

Practice

Considerf(x) =x^3.

1. Sketchf(x)andf−^1 (x).


2. Findf−^1 (x)algebraically. It is actually a function?


3. Verify algebraically thatf(x)andf−^1 (x)are inverses.
   Considerg(x) =√x.


4. Sketchg(x)andg−^1 (x).


5. Findg−^1 (x)algebraically. It is actually a function?


6. Verify algebraically thatg(x)andg−^1 (x)are inverses.
   Considerh(x) =|x|.