Questions

$$\text{ Is } y = x^3 \text{ an odd function, an even function, or neither?}$$

$$\text{ Is } y = x^4 \text{ odd, even, or neither?}$$

$$\text{ Is } y = x^3 + x \text{ odd, even, or neither?}$$

$$\text{ Is } y = x^3 + x^2 \text{ odd, even, or neither?}$$

$$\text{ Is } y = |x| \text{ odd, even, or neither?}$$

$$\text{ Is } y = \sin x \text{ odd, even, or neither?}$$

$$\text{ Is } y = \cos x \text{ odd, even, or neither?}$$

$$\text{ Is } y = e^{x} + e^{-x} \text{ odd, even, or neither?}$$

$$\text{ Is } y = e^{x} – e^{-x} \text{ odd, even, or neither?}$$

Solutions

1)
Even functions have the property f(x) = f(-x)

Odd functions have the property f(-x) = -f(x)

$$(-x)^3 = -x^3$$ $$\text{ Therefore } y=x^3 \text{ is an odd function.}$$

2)

$$(-x)^4 = x^4 \Rightarrow \text{ even.}$$

3)
Both $x^3$ and $x$ are odd. Sum of odd functions is odd.

$$\text{ odd.}$$

4)
$x^3$ is odd, $x^2$ is even. Sum of an odd and an even function is neither.

$$\text{ neither.}$$

5)

$$|-x| = |x| \Rightarrow \text{ even.}$$

6)

$$\sin(-x) = -\sin x \Rightarrow \text{ odd.}$$

7)

$$\cos(-x) = \cos x \Rightarrow \text{ even.}$$

8)

$$(e^{x} + e^{-x})\text{ at }-x:\ e^{-x} + e^{x} = e^{x}+e^{-x} \Rightarrow \text{ even.}$$

9)

$$(e^{x} – e^{-x})\text{ at }-x:\ e^{-x} – e^{x} = -(e^{x}-e^{-x}) \Rightarrow \text{ odd.}$$