Classify each function as even, odd, or neither even nor odd: f(x) = 4x^3 - 3x^2 + 5, f(x) = -7x^2 + 1, y = cos(-x), y = sin x / cos x, y = csc x, and y = tan x / x.

Step-by-step solution

1. 1

   Fix the test. $f$ is even when $f(-x)=f(x)$ and odd when $f(-x)=-f(x)$. For polynomials there is a shortcut: only even powers means even, only odd powers means odd, a mixture means neither. (A constant counts as $x^0$, an even power.)

2. 2

   Sort the two polynomials. $4x^{3}-3x^{2}+5$ contains the odd power $x^3$ and the even powers $x^2$ and $x^0$, so it is neither. In $-7x^{2}+1$ every power is even, so it is even.

3. 3

   Simplify cos(-x) before classifying. Cosine is even, so $\cos(-x)=\cos x$ — the expression is just the cosine function in disguise, and it is even. Testing directly: replacing $x$ by $-x$ gives $\cos(x)$, which equals the original.

4. 4

   Classify sin x / cos x. This is $\tan x$. With $\sin(-x)=-\sin x$ and $\cos(-x)=\cos x$:

   $\frac{\sin(-x)}{\cos(-x)}=\frac{-\sin x}{\cos x}=-\tan x,$

   so it is odd.

5. 5

   Classify csc x. Since $\csc x=\dfrac{1}{\sin x}$ and sine is odd,

   $\csc(-x)=\frac{1}{-\sin x}=-\csc x,$

   so it is odd. The reciprocal of an odd function is odd.

6. 6

   Classify tan x / x. Tangent is odd and $x$ is odd, and odd divided by odd is even:

   $\frac{\tan(-x)}{-x}=\frac{-\tan x}{-x}=\frac{\tan x}{x},$

   so it is even. Final sorting — even: $-7x^2+1$, $\cos(-x)$, $\tfrac{\tan x}{x}$; odd: $\tfrac{\sin x}{\cos x}$, $\csc x$; neither: $4x^3-3x^2+5$.