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

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

58 STEP 4. Review the Knowledge You Need to Score High

− 33 x

y

0

3

Figure 5.3-1

Since the graph ofy=

√

9 −x^2 passes the vertical line test, the equation is a function.

Lety= f(x). The expression

√

9 −x^2 implies that 9−x^2 ≥0. By inspection, note that

− 3 ≤x≤3. Thus, the domain is [−3, 3]. Sincef(x) is defined for all values ofx∈[−3, 3]

andf(−3)=0 is the minimum value and f(0)=3 is the maximum value, the range off(x)

is [0, 3].

Example 2

Givenf(x)=x^2 − 4 x, findf(−3), f(−x), and

f(x+h)−f(x)

h

.

f(−3)=(−3)^2 −4(−3)=^9 +^12 =^21

f(−x)=(−x)^2 −4(−x)=x^2 + 4 x

f(x+h)−f(x)

h

=

(x+h)^2 −4(x+h)−(x^2 − 4 x)

h

=

x^2 + 2 hx+h^2 − 4 x− 4 h−x^2 + 4 x

h

=

2 hx+h^2 − 4 h

h

= 2 x+h−4.

Operations on Functions

Let f andg be two given functions. Then for allxin the intersection of the domains of

f andg, thesum,difference,product, andquotientof f andg, respectively, are defined as

follows:

(f+g)(x)=f(x)+g(x)

(f−g)(x)=f(x)−g(x)

(fg)(x)=f(x)−g(x)

(

f

g

)

(x)=

f(x)

g(x)

,g(x)= 0

The composition offwithgis (f◦g)(x)= f(g(x)), where the domain off◦gis the set

containing allxin the domain ofgfor whichg(x) is in the domain off.

Example 1

Givenf(x)=x^2 −4 andg(x)=x−5, find

(a) (f◦g)(−1)

(b) (g◦f)(−1)

(c) (f+g)(−3)