0
votes
3answers
38 views

How do I show algebraically that the period of the tangent function is $\pi$?

How do I show that the positive real number $p$ for which $\tan (x+p)=\tan (x)$ is equal to $\pi$? In essence how do I prove the period of the tangent function is $\pi$? Please bear in mind I am a ...
-1
votes
3answers
56 views

Does f(x) = g(u)?

If $f(x) = x + \sqrt{2-x}$ and $g(u) = u + \sqrt{2-u}$ is it true that $f = g$? I squared both sides $\sqrt{x + \sqrt {2-x}} = \sqrt{u + \sqrt {2-u}}$ $\sqrt{x} + 2-x = \sqrt{u} + 2-u$ I then ...
0
votes
2answers
69 views

Show that $f:\mathbb{R}-\{2\}\to\mathbb{R}-\{5\}$ with $f(x)=\frac{5x-1}{x-2}$ is bijective

Can anyone please help to explain the question and what actually $f: \mathbb{R} - \{2\}$ means ?? I know that bijection means one to one function and onto both. Any idea to start up with this ...
0
votes
1answer
19 views

Finding the vertical shift of a sinusoidal function

I'm currently studying sinusoids, I've been given a graph with a few key points and have been told to find a cosine function which fits it. When it comes to finding the vertical shift of the graph the ...
-3
votes
2answers
62 views

Clarification of Identification [on hold]

This is more of an observation question. When you see $x$, In $f(x) = x^2$ And when you see $g(x) = x^3$ You automatically identify $x = x$ Wouldn't the $x$'s be off by a little bit? But ...
0
votes
2answers
27 views

Simplify a function with a square root as the numerator

How would I go about simplifying this: $$\frac{\sqrt{x^4 + 3x^2}}{x}$$ thanks!
3
votes
0answers
36 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
0
votes
1answer
27 views

Is there a way to do this? Fixed deduction for x rounds where total = fixed amount

I am trying to calculate the reduction amount / step per round for the given: rounds = 1000 points = 80 starting at reward = 1 point So from round 1 which has a reward of 1 point deduct a fixed ...
1
vote
2answers
39 views

How to find the equation of the graph reflected about a line?

Consider the graph of $y = e^x$ (a) Find the equation of the graph that results from reflecting about the line $y = 4$. (b) Find the equation of the graph that results from reflecting ...
0
votes
1answer
19 views

Quadratic Functions: Determine the value of b

I'm having trouble with this question and I'm not sure what to do. Would appreciate any one who helps me out. Question: The point $(-2,1)$ is on the graph of the quadratic function: $f(x) = -x^2 + bx ...
0
votes
1answer
46 views

How to prove: if $x = y$, then $f(x) = f(y)$ (for a function $f$)

The intuition of this for me is that if two elements in the domain are equal, then their images are equal. But I don't know if this is actually true or not, so I want to know if this is for sure. ...
2
votes
2answers
72 views

Why can't equations with unknown inside and outside of a function be solved in a standard way?

For example the equation $$ n2^n = 8 $$ Is true for $n=2$ which can be guessed, but more complicated examples would require some sort of approach. Also with trigonometric functions, $$ x\sin(x) + ...
0
votes
2answers
56 views

Not understanding the answer to the inverse of a function

$$f(x)\quad =1+\sqrt { 1+x } $$ $$y\quad =1+\sqrt { 1+x } $$ $$y^{ 2 }\quad =1+1+x$$ $$y^{ 2 }-2\quad =x$$ How is it $x=y^{ 2 }-2y$ ?
0
votes
2answers
34 views

Finding asymptotes for $f(x)=\frac{x^2+3x-10}{3x^2+13x-10}$

$$f(x)=\frac{x^2+3x-10}{3x^2+13x-10}$$ I know that the horizontal asymptote is $1/3$. To find the vertical asymptotes, I set the denominator equal to zero and used the quadratic formula, and I got ...
0
votes
0answers
10 views

Give the transformations of the following functions.

Give the transformations of the following 3 functions. Can you please give me at least 3 points to plot for each function(keeping the domain restriction in mind)? Also for rational function. Also ...
0
votes
2answers
20 views

Compositions with Restricted Domain.

Hey Guys! How do you do this problem. There is no overlapping domain for f(x) and g(f(x)). Detailed explanation as to how to do the problem will be preferred! Thanks.
1
vote
1answer
23 views

Compositions and Restricted Domains

Hey guys. I got the answer under question 3 (which is circled). Can you please verify if it is correct? If not, can you please specify how to go about this problem? Thanks
0
votes
2answers
45 views

How to find the composition of two functions and its domain?

I have no clue how to go about this problem. A detailed explanation would be preferred. Thanks
0
votes
1answer
55 views

How to find the domains of functions $f(x) = x-5$, $g(x) = \sqrt{x-5}$, and of their sim?

I've been studying on Study Plan Practice, on MyMathLab for my College Algebra class. We're going over the Algebra of Functions right now and several things don't make much sense. The question is: ...
0
votes
1answer
79 views

$f(x+h)$ not equal to $f(x) +f(h)$???

I'm taking College Algebra at a local community college, and I just wasn't able to follow how my professor came to these conclusions. (3 separate times.) $$\frac{f(x+h) - f(x)}{h},$$ $$f(x) = ...
1
vote
1answer
35 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
2
votes
4answers
79 views

Showing Surjectivitity of $f(x) = x^3$

I want to show that the function $f: \mathbb{R} \to \mathbb{R},\; f(x) = x^3$ is surjective. First Question: If a function has an inverse, it is bijective yes? Second Question: Is my process ...
-2
votes
2answers
39 views

Hey guys. Given the graph below, find the equation of the transformed parent function. [closed]

It would be great if there is a detailed explanation. Also, is there a standard method I can use to answer all kinds of graphs including exponents and logs? Thanks
-1
votes
2answers
53 views

Proof of period of $f(ax+b)$

I have been taught that $f(x)$ is called a periodic function with period $T$ if $$f(x)=f(x+T)$$ This I understand completely. Also I have been taught that $$f(ax)=f(ax+T/|a|)$$ if $f(x)$ has a period ...
4
votes
2answers
53 views

Piecewise linear function and absolute value

While writing a solution to homeworks for my students, I had to write the function $$f(x)=\left\{\begin{array}{ll} \frac{x+2}{2}, & x\leqslant -4\\ \frac{x}{4}, & -4\leqslant x\leqslant 4 \\ ...
4
votes
1answer
65 views

Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$?

The exercise asks me this: Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$? ps: $f: \mathbb{R}\to \mathbb{R}$ I really don't know how to start :c, I appreciate hints.
16
votes
3answers
236 views

Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

My idea is that the two functions are not the same since for the first function, the domain of the function is only non negative reals for the numerator and positive reals for the denominator. ...
11
votes
3answers
278 views

What happens to a function when it is undefined?

If I have the function $$f(x) = {x^2 - 2 \over x + \sqrt 2}$$ this is undefined for $x = -\sqrt 2$, am I correct? Since the denominator would be zero. But the numerator is a difference of ...
3
votes
2answers
63 views

Expressing the area as a function :)

Express the area A of an equilateral triangle as a function of the height of the triangle. Thanks :) I am not sure where to even start on how to answer this problem.
1
vote
2answers
80 views

How to find the domain and range of $f(x) = \sqrt{x^2-2x+5}$?

This is the function: $$f(x) = \sqrt{x^2-2x+5}$$ Edit: normally what I would do is this: Since it's a square root function, the thing inside the root has to be $\ge 0$. So, $(x^2 - 2x+5)\ge 0$. Then ...
5
votes
2answers
78 views

What is this notation for a function? I've never seen it written like this before.

What does this mean? $$ f=\{ (x,y): y= x+2 \}$$ I don't understand what "$(x,y):$" means in regard to the problem. Also why is the $y$ inside of the $f(x)$ function. Isn't it supposed to be outside? ...
1
vote
2answers
31 views

Sketching graphs abs value functions

how do I go forward with sketching the graphs of the following two functions? i) $y(t)=|2+t^3|$ ii) $f(x)=4x+|4x-1|$ thanks in advance!
0
votes
4answers
59 views

Find domain and range of $(f \circ g)$ for $f(x)=\ln x$ and $g(x)=x^2−1$

Word for word: Consider the functions $f(x)=\ln x$ and $g(x)=x^2−1$, find the domain and range of $(f\circ g)(x)$ I think this is asking to find the domain and range of $\ln(x)^{2}-1$ and the ...
3
votes
1answer
107 views

Solve for $f(x)$ if $f(f(x))=6x-f(x)$

If $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)
5
votes
4answers
706 views

generalized way of finding minimum value of a function?

$f(x)=\frac{x^{2}-1}{x^{2}+1}$ for every real number for $x$, the minimum value of $f$ is what? How can I find the minimum value of this function.I only know trial and error method, but it's not a ...
1
vote
1answer
88 views

How to graph $y=f(x^2)=\sin(x^2)$?

How to graph $y=f(x^2)=\sin(x^2)$? I have substituted as follows: $$\begin{cases} y=f(a)=\sin a\\ a=x^2\end{cases}.$$ Then if I graph this with the coordinate axes $y$ and $a$ I get the ordinary ...
2
votes
4answers
566 views

How do I show that f is strictly decreasing on (0, infinity)?

I have been asked to define $f: (0, \infty) \to (0, \infty)$ by $f(x) = \frac 1 x$ a) How do I show that f is strictly decreasing on $(0, \infty)$? I realize that I have to show that $f'(x)<0$, ...
2
votes
1answer
60 views

Injective map from real projective plane to $\Bbb{R}^4$

Consider the mapping $\Bbb R^3\rightarrow\Bbb R^4$ given by $$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$ which passes to the quotient and can therefore be viewed as a map from the projective plane $\Bbb ...
7
votes
1answer
202 views

finding the value of $f(2001) $ if…

if $f (\frac{x}{y}) =\frac{f(x)}{y} $ and $f(2000)=1$ ; then what's the value of $f(2001)$. I tried hard but can't figured out anything. please help me, how can I proceed?
0
votes
2answers
63 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
2
votes
3answers
35 views

How do I find the domain of this function

I would like to know which operations i have to do to get the domain of this function: $$y=\sqrt{\frac{1}{x}-1}$$ I have researched and the solution of the inequality $\frac{1}{x}-1 \geq 0$ is ...
1
vote
2answers
56 views

Graph a function

I have a question, I have a function: $$f(x) = \frac{-x^2-10x}{2}$$ I'm really confused how to replace the x. So, what would be the points in $y$ if $x$ were: $-4, -3, -2, -1, 0, 1, 2, 3, 4$?
3
votes
2answers
52 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
0
votes
1answer
22 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
5
votes
2answers
175 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
-1
votes
1answer
43 views

Why does $f(x) = x \cos 4x$ lie below the $x$-axis in ther interval $[\pi/4, \pi/3]$?

How can I explain why function $f(x)$ lies below the $x$-axis in the interval $\left[\dfrac{\pi} 4, \dfrac{\pi} 3\right]$ where $f(x) = x \cos 4x$?
0
votes
1answer
24 views

Graphing functions

I am having problems understanding how to graph the product $fg$ when $f(x) = x$ and $g(x) = |x|$. Any help would be much appreciated!
2
votes
1answer
39 views

Greatest value of f

If $f'(x)=6-x$ then which of the following has the greatest value? $f(2.01)-f(2)$ $f(3.01)-f(3)$ $f(4.01)-f(4)$ $f(5.01)-f(5)$ $f(6.01)-f(6)$ I know the answer is $f(2.01)-f(2)$ but how to prove?
6
votes
1answer
69 views

What am I doing wrong in this algebra excercise?

This is my first question here, so please forgive me if the format etc. are not quite right. I've been attacking an algebra question, and my workings are below. There's a mistake somewhere (I don't ...
-1
votes
1answer
31 views

What is the domain? [closed]

What is the domain of the function $f(x) = \sqrt{4 x + 37}$? I am not sure where to get started with this problem. Thanks in advance!