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I have a number of exercises (some examples below) that ask to determine some properties of functions:

  • Symmetry with respect to y-axis or origin. (I know how to make a graph, but for some graphs, this can be confusing)
  • Even or Odd function, meaning $f(-x) = f(x)$ or $(f-x) = -f(x)$. (No idea, never learned this.)
  • Is the function periodic? If so, state the period. (I guess it means if it's contained within one period?)
  • Is $f(x)$ a one-to-one function? (For each $f(x)$ only one $x$ exists). I have no clue about this one.

What should I do answer these?

For example, I graphed function (l) below but I do not understand the graph. On WolframAlpha with the function of l, it said it is an infinite cone but I have yet to graph a 3D figure. So any explanation on that would be helpful.

The example functions are:
f) $f(x)=\tan(x)$
g) $f(x)=\sec(x)$
h) $f(x)=2^x$
i) $f(x)=\log_2x$
l) $f(x)=\sqrt{a^2-x^2}$ (assuming, a=1)

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    @AustinBroussard symmetry across the y-axis is the same as $f(x)=f(-x)$. As in "if you're at a position, flipping across the y-axis (which is $-x$) keeps you at the same height". Odd means "if you're at a position, flipping around the y-axis *and* x-axis puts you on the function again" $f(-x)=-f(x)\implies -f(-x)=f(x)$. the inner negative is the same y-axis flip, the outer one is the x-axis flip.2012-07-15

1 Answers 1

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A function $f$ is said to be "even" if and only if for every $x$ in the domain, $f(x)=f(-x)$. The simplest way to check this is to verify equality after you plug in. To verify that a function is not "even" we need to exhibit a particular number $x$ for which $f(x)$ and $f(-x)$ are distinct.

For instance, with $f(x)=\tan(x)$, $f(x)$ is not even: if $x=\frac{\pi}{4}$, then $f(x) = f\left(\frac{\pi}{4}\right) = 1$ but $f(-x) = f\left(-\frac{\pi}{4}\right) = -1 \neq f(x).$

On the other hand, $f(x) = \sqrt{a^2-x^2}$ is even: the domain is $[-|a|,|a|]$ (since we are assuming $a=1$, this would be $[-1,1]$. If we plug in $-x$ instead of $x$, we have: $f(-x) = \sqrt{a^2 - (-x)^2} = \sqrt{a^2 - x^2} = f(x),$ and this holds for all $x$, so $f$ is even.

Check the other functions; (by the way, if you have values of $x$ that are in the domain but $-x$ is not in the domain, then the function is not even.)

"Odd" is similar, except that the definition is that for all $x$ in the domain, we must have $f(-x) = -f(x)$.

For example, $f(x)=\tan(x)$ is odd, since for all $x$ in the domain, we have $f(-x) = \tan(-x) = \frac{\sin(-x)}{\cos(-x)} = \frac{-\sin(x)}{\cos(x)} = -\frac{\sin(x)}{\cos(x)} = -\tan(x) = -f(x).$ (Using the fact that $\sin(x)$ is odd and $\cos(x)$ is even).

A function $f$ is periodic if there exists $p\gt 0$ such that $f(x)=f(x+p)$ for all $x$. For example, $\sin(x)$ is periodic, since $\sin(x+2\pi) = \sin(x)$ for all $x$. On the other hand, $f(x) = 2^x$ is not periodic, since we know that for all real numbers $a$ and $b$, if $a\lt b$ then $2^a\lt 2^b$, so we can never "repeat values". The number $p$ would be a period of the function. (Generally, there are many periods, since if $p$ is a period, then so is $2p$, and $3p$, and $4p$, etc.

A function $f(x)$ is one to one if $a\neq b$ implies $f(a)\neq f(b)$ (equivalently, if whenever $f(a)=f(b)$, then it must be the case that $a=b$). You can prove that a function is not one-to-one by exhibiting a single pair of numbers $a\neq b$ where the function takes the same value. For example, $f(x)=x^2$ is not one-to-one, because even though $1\neq -1$, we nevertheless have $f(1)=f(-1)$. On the other hand, $f(x) = x^3$ is one-to-one, because if $f(a)=f(b)$, then that means that $a^3=b^3$, and the only way this can occur is if $a=b$.

All of this is algebraically.

Geometrically, assuming you can get nice and accurate graphs, a function $f(x)$ is even if the graph of $y=f(x)$ is symmetric about the $y$ axis; the function is odd if it is symmetric about the origin. It is periodic if it "repeats" after a finite length (think about the graph of $y=\sin(x)$). And it is one-to-one if it passes the "horizontal line test":

Horizontal Line Test The graph of $y=f(x)$ intersects each horizontal line in at most one point.