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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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closed as too localized by J.D., Norbert, William, Henning Makholm, Asaf Karagila Aug 4 '12 at 15:01

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

This is something a little too long. Maybe you can start with some and see how to generalize. – Pedro Tamaroff Jul 14 '12 at 23:02
the key point on l) is that a=1. Set that, then do the resulting graph (that only involves $x$). even/odd and symmetry around the y-axis/origin are the same, which should help. one-to-one means that any $y$ value is only taken once. That is, one x-value per one y-value. For example, $x^2$ is not one to one, since $2$ and $-2$ both end up at $4$. A better question may be for an explanation of these terms rather than a (quite large) list of questions. You'll get used to the flow of questions here with time. – Robert Mastragostino Jul 14 '12 at 23:03
Austin, you are committing a breach of etiquette; the downvotes should indicate to you the same. Ask one or two pieces at at time. Do not phrase your inquiry in the form of an imperative. Indicate what you have done to attempt solve the problem. Use the lingua franca of, LaTeX formatting. – ncmathsadist Jul 14 '12 at 23:04
@AustinBroussard: Since you are asking about twelvel functions, perhaps you can pick two or three of those (say, representing polynomials, trig functions, and "others"), and ask about those only. It is also helpful if you tell us what you have been able to do and how. Then we can walk you through a couple of examples, and then you can try the rest on your own. Please consider doing that. – Arturo Magidin Jul 14 '12 at 23:04
Like I said. New to the site. But if these could be brought up slightly. I would just like someone to describe what the top is and I'll see if I can figure out how to do them and I will edit this post with my attempts. Thank you all for bearing with me. – Austin Broussard Jul 14 '12 at 23:07
up vote 1 down vote accepted

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.

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