Functions in Algebra Examples

Is it true that f(n)=O(g(n)) implies g(n)=O(f(n))

It is function is odd function?
$y=2x^3-<!--\; -\; -->3x^2$

Which statement is true for the function $f(x)=\ln <!--\; \; -->(\tan <!--\; \; -->x)$ on the openinterval $\mathrm{\pi <!--\; \pi \; -->}<x<\frac{5\mathrm{\pi <!--\; \pi \; -->}}{4}$?
a. f(x) is increasing at an increasing rate.
b. f(x) is increasing at a decreasing rate.
c. f(x) has a point of symmetry in the open interval.
d. f(x) has an absolute maximum in the open interval
e. f(x) has a point of inflection in the open interval.

How to find period of a real function f given the functional equation $\sqrt{3}f(x)=f(x-<!--\; -\; -->1)+f(x+1)$?
If a periodic function satisfies the equation $\sqrt{3}f(x)=f(x-<!--\; -\; -->1)+f(x+1)$ for all real x then prove that fundamental period of the function is 12.
Here fundamental period means the smallest positive real for which function repeats its value for all x.
I tried replacing x by $x\pm <!--\; \pm \; -->1$ then try to find f(x) in terms of other but always end up with it in terms of sum of other two arguments in the function eg $f(x+2)$ etc.
Please provide a general method and also especially do give the thought process or reasoning for all the steps ie why you are doing these particular steps or what led you to thinking that doing these steps would give you the period of f.

a)Apply the chain rule to u = xyz, where x, y, and z are functions of t, to get a rule for differentiating a product of three functions of one variable.
b)Derive the rule in (a) by using one-variable calculus.

Find the absolute minimum value of $f(x)=5+4x+\frac{16}{x}$ for x>0

Divide $\frac{(x^3+x-<!--\; -\; -->30)}{x-<!--\; -\; -->3}$

What is the difference between a formula and a function?

Which rational functions is graphed below

Which of the following statements is true for the function
$f(x)=\frac{x^2+x}{2(x^2-<!--\; -\; -->1)}$
a) y=1^2 is a horizontal asymptote and x=2 is a vertical asymptote.
b) y=1^2 is a horizontal asymptote and x=1 is a vertical asymptote.
c) y=3 is a horizontal asymptote and x=-1 is a vertical asymptote.
d) y=3 is a horizontal asymptote and x=2 is a vertical asymptote

Find the global maximum and minimum for the function on the closed interval. Show your work and justify your answer. A calculator answer is not acceptable.
$f(x)=x-<!--\; -\; -->2\ln <!--\; \; -->(x+1),\text{ where }0\le <!--\; \le \; -->x\le <!--\; \le \; -->2$

For the function $32/x+3$ , find $(g-<!--\; -\; -->1?g)(5).$

Which set of ordered pairs in the form (x,y) does not represent a function of x?
A. $(-<!--\; -\; -->1,2),(3,-<!--\; -\; -->2),(0,1),(5,2)$
B.$(-<!--\; -\; -->1,2),(3,2),(-<!--\; -\; -->2,2),(0,2)$
C.$(-<!--\; -\; -->1,2),(3,-<!--\; -\; -->2),(0,1),(3,5)$
D.$(-<!--\; -\; -->1,2),(2,3),(3,-<!--\; -\; -->2),(-<!--\; -\; -->2,0)$

What is the measure of a straight line angle?
A.90 degrees
B.360 degrees
C.270 degrees
D.180 degrees

The function $f(x)=x^2-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}\le <!--\; \le \; -->x\le <!--\; \le \; -->\mathrm{\pi <!--\; \pi \; -->}$ is:

Which of the following is not true?
A. Every relation from the set A to the set B is a function from the set A to the set B.
B. If f is a function from A to B and (a, b) Є f, then f(a) = b, where a is called the pre-image of b under f.
C. Every function from the set A to the set B is a relation from the set A to the set B
D. If f is a function from A to B and (a, b) Є f, then f(a) = b, where b is called the image of a under f.

What is the derivative of $2^x$?

What is the slope of the line represented in the graph?

$\cos ({\tan }^{-1}x)=?$
A$1+x^2$
B$\frac{1}{1}$
C$1+x^2$
D None of these