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

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

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

Example 1

f(x)=x^4 −x^2

Begin by examiningf(−x). Sincef(−x)=(−x)^4 −(−x)^2 =x^4 −x^2 ,f(−x)=f(x). Therefore,

f(x)=x^4 −x^2 is an even function. Or, using your graphing calculator, you see that the

graph off(x) is symmetrical with respect to they-axis. Thus,f(x) is an even function. Or,

sincefhas only even powers, it is an even function. (See Figure 5.4-10.)

[−4, 4] by [−3, 3]

Figure 5.4-10

Example 2

g(x)=x^3 +x

Examineg(−x). Note thatg(−x)=(−x)^3 +(−x)=−x^3 −x=−g(x).Therefore,g(x)=

x^3 +xis an odd function. Or, looking at the graph ofg(x) in your calculator, you see that the

graph is symmetrical with respect to the origin. Therefore,g(x) is an odd function. Or, since

g(x) has only odd powers and a zero constant, it is an odd function. (See Figure 5.4-11.)

[−4, 4] by [−3, 3]

Figure 5.4-11

Example 3

h(x)=x^3 + 1

Examineh(−x). Sinceh(−x)=(−x)^3 + 1 =−x^3 +1,h(−x)=h(x), which indicates thath(x)

is not even. Also,−h(x)=−x^3 −1; therefore,h(−x)=−h(x), which implies thath(x)isnot

odd. Using your calculator, you notice that the graph ofh(x) is not symmetrical with respect

to they-axis or the origin. Thus,h(x) is neither even nor odd. (See Figure 5.4-12.)

[−4, 4] by [−3, 3]

Figure 5.4-12