CPM Homework Help : CALC Problem 2-125

Determine if the following functions are even, odd, or neither. Explain how you determined your choice.

Hint:

$y=\sin^2x$

Step 1(a):

$y=\sin^2x=\left(\sin x\right)\left(\sin x\right)$

Hint (a):

$y=\sin x$ is an odd function. Definition of odd functions: $f\left(−x\right) = −f\left(x\right)$ Therefore, $\sin\left(−x\right)=−\sin\left(x\right)$ .

Step 2(a):

Explore what $y=\sin^2\left(−x\right)$ looks like:
$y=\sin^2\left(−x\right)$
$=\left(\sin\left(−x\right)\right)\left(\sin\left(−x\right)\right)$
$=\left(−\sin x\right)\left(−\sin x\right)$
$=\sin^2x$

Answer (a):

Hence $\sin^2\left(−x\right)=\sin^2\left(x\right)$ .
This is the definition of an even function: $f\left(−x\right) = f\left(x\right)$ .

$y = \frac { x ^ { 2 } + 1 } { x ^ { 3 } - 2 x }$

Step 1(b):

Test for even: $f\left(a\right) = f\left(−a\right)$

$f(a)=\frac{a^{2}+1}{a^{3}-2a}$

$f(-a)=\frac{(-a)^{2}+1}{(-a)^{3}-2(-a)}=\frac{a^{2}+1}{-a^{3}+2a}\ ,$

$f\left(a\right) ≠ f\left(−a\right)$ ; not even

Step 2(b):

Test for odd: $f\left(−a\right) = −f\left(a\right)...$