Answer to a Function Question

Question

Hello I am new to functions and I'm unsure on two questions.

"Let the function be defined by

f(x) = 2x² - 2 for for 0 ≤ x <3

= 28 - 4x for 3 ≤ x < 6

= f (x + 6) for all x

Find the values of f(0), f(2), f(5), f(38), and the period of the function.

I know the answers to 0, 2, and 5, but I don't know how to get the answer to the function if it's 38, and also I don't know what the period of the function is? Help would be great! Thank you!

f(0) = -2

f(2) = 6

f(5) = 8

f(38)= ??

period = ??

Answer 1

Functions that repeat themselves regularly are called periodic. Repeating means that if the x value is increased by some constant value, the y value remains the same. Mathematically, this means that f(x+k) = f(x) for all values of x for some constant k. In particular, if f(x) = f(x+6) for all x, then the period of f is 6.

Note that if the function repeats every time x is increased by 6, then it will also repeat every time the function is increased by 12, or 18, or any other multiple of 6. This means that

f(x) = f(x+6) = f(x+12) = f(x+18) = f(x+24) = f(x+30) = f(x+36)

To get f(38), set x = 2 to find f(2) = f(38)