i am having trouble filling out the chart of basic functions that includes periodic and one by one
i dont understand the concept
X^2
X^3
Abs(x)
Sin(x)
Cos(x)
Tan(x)
Sec(x)
2^x
Logbase 2 of x
1/x
Sq rt of X
sq rt of a^2- x^2
i suppose you mean one to one?
intuitively, a function is periodic if it has a repeating pattern, forever. sin(x) is obviously one. it's period is $\displaystyle 2 \pi$. you have the same pattern between $\displaystyle [0, 2 \pi]$ repeating forever.
slightly more formally, a function $\displaystyle f(x)$ is called periodic, if $\displaystyle f(x) = f(x + kT)$ for some integer $\displaystyle k$. we call $\displaystyle T$ the period.
for example, going back to sine. $\displaystyle \sin x = \sin (x + 2k \pi)$, the period is $\displaystyle T = 2 \pi$. any multiple of the period and you get back the same value. so $\displaystyle \sin x = \sin (x + 2 \pi) = \sin (x + 6 \pi) = \sin (x - 12 \pi)$ etc
a function is called one-to-one if $\displaystyle f(x_1) = f(x_2) \implies x_1 = x_2$. or equivalently, $\displaystyle x_1 \ne x_2 \implies f(x_1) \ne f(x_2)$ for $\displaystyle x_1, x_2 \in \text{dom}(f)$