math problem I dont get

Question

Determine if the following functions are one-to-one, even,odd, periodic

(a) f(x)=x²+6x−8, for x∈[3,∞)

b) f(x)=tan (7x) for x∈{-pi,pi}

c) f(x)=cos(9x) for x∈R

d) f(x)=8x^5

please help me I dont get this

Answer 1

The easiest way of checking one-to-one is by using the horizontal line test. Graph the function on the given interval. If there is a horizontal line intersecting the graph in more than one point, the function is not one-to-one.

To check even/odd, replace x with -x and see if you get back f(x), -f(x), or neither. In the first example,

f(-x) = (-x)^2 + 6(-x) - 8 = x^2 - 6x - 8,

which is not f(x) or -f(x), so it is neither even nor odd. For part (d), f(-x) = 8(-x)^5 = -8x^5 = -f(x), so this f is odd.

I would check periodic with the graph: does the graph repeat itself over some interval? Functions (b) and (c) are periodic, but the others are not.