0
votes
2answers
56 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
34 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
54 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
42 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
20 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
158 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
37 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
22 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 ...
0
votes
1answer
30 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!
0
votes
0answers
5 views

tangent vector on cone

Suppose that $C$ is a cone with vertx at the origin and let $\nu(x)= (\nu_1(x),\cdots, \nu_n(x))$ is the tangent vector at the point $x \in \partial E$. Is $\nu(x)$, (as function of $ x \in \partial ...
3
votes
3answers
42 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...
0
votes
3answers
82 views

If $H = \{(1,-4),(2,-3),(3,-2),(4,-1),…\},\;$ what is $H(9)$? [closed]

If $H = \{(1,-4),(2,-3),(3,-2),(4,-1),...\}$, what is $H(9)$? This is the problem on our lesson about relation and functions. Please help.
0
votes
1answer
40 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
1
vote
3answers
62 views

Show that $x^3 +x-1$ has a zero between $x=0$ and $x=1$

Show that $x^3 +x-1$ has a zero between $x=0$ and $x=1$, does anyone know how to go about starting this problem? I am basically clueless. I thought maybe at first polynomial division since its $x^3$, ...
3
votes
0answers
56 views

Functions that are defined by the equation [closed]

How many different functions of $x$ are defined by the equation $x^2+y^2=9$ if the domain is $x\in [-2,2]$? (A) None (B) 1 (C) 2 (D) 4 Need help finding out how many functions ...
1
vote
5answers
54 views

Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
0
votes
1answer
34 views

Finding the range and domain of $h(x) = \sec (x)$

I am attempting to show how to find the range and domain of $h(x) = \sec (x)$. Here's my working so far. Consider $h(x) = \sec (x)$, which is defined as $h(x) = \sec (x)=\frac{1}{\cos(x)}$. We know ...
1
vote
3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
2
votes
3answers
242 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
5
votes
1answer
77 views

Find polynomials f(x),g(x) and h(x),if they exist,such that for all x… Putnam Problem

Find polynomials $f(x), g(x)$, and $h(x)$, if they exist, such that for all $x$, $\mid f(x)\mid-\mid g(x) \mid+h(x)= \begin{cases} -1, & \text{if}~x<-1 \\ 3x+2, & \text{if}~-1\leq ...
0
votes
2answers
34 views

PreCalculus Domain of a function

What is the domain of $(x-3)^{1/2}\;?$ The domain would be $x>3$ wouldn't it? Because the graph doesn't exist until after the point $(3,0)$
0
votes
1answer
31 views

PreCalculus Composition of Functions

If $h(x) = ((x - 2)^2)$, find $h(x + 1)$, $$=((x+1)-2)^2 =(x-1)(x-1) =x^2-2x+1$$. I'm certain I've made a mistake in here
0
votes
1answer
45 views

How can I find the value of $c$ given an odd function with these intervals?

An odd function which is defined for all real numbers $x$ has these intervals: $(-3, 5)$, $(-2, -4)$, $(2, c)$. How can I find the value of $c$? I have no idea on this one.
0
votes
4answers
39 views

Finding the three unknowns

Can someone show me the steps to finding the three unknowns of these two equations. $$-a-bx+cx^2 = x^2+2x+1$$ The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they ...
1
vote
1answer
45 views

roots of sum of exponential functions

Could anyone point me in the right direction of finding the roots of equations of the form $$ \sum_{i=1}^n a_ie^{f_i(x)}, $$ where $a_i \in \mathbb{R}$ and the $f_i$ are each first degree polynomials ...
-1
votes
1answer
30 views

Show the image of this function

Show that the image of the function (no derivative) $$f(x)=\frac{x^2+x+1}{x}$$ is: $$(-\infty,-1]\cup[3,\infty)$$ I tried to prove that it was increasing range and decreasing in others (maximum ...
1
vote
2answers
50 views

How does Wolfram get from the first form to the second alternate form?

So, I was trying to compute an integral but I couldn't actually manage getting anywhere with it in its initial form. So, I inserted the function in Wolfram Alpha and I really got a nicer form (second ...
0
votes
2answers
49 views

Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
2
votes
3answers
36 views

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively…

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively. Find the remainder when P(x) is divided by $(x-a)(x-b)(x-c)$ is (a,b,c are distinct) My ...
0
votes
1answer
28 views

Determining odd or even functions

I know that to determine wether or not a function is even, you sub in $(-x)$ for $x$, and see if it the same (even) or not (odd/neither). However, I get confused when you have to sub in $(-x)$ for ...
2
votes
3answers
103 views

Solutions of $x^2-6x+\lfloor x \rfloor+7=0$

What are the roots of $x^2-6x+\lfloor x \rfloor+7=0$, where $\lfloor x \rfloor$ is the greatest integer function? Is there some way to solve the equation without graphing?
0
votes
0answers
28 views

Graphing inequalities, elementary geometry (Highschool level)

I am going through some past papers, and one of the few questions I need help with are: For the second one, relating to plotting the square root of 4 - $x^2$, I did a table of values starting ...
0
votes
1answer
16 views

Continuous function cuts y=x line

Well this might be very trivial or just an application of a theorem unknown to me - If $(a,b)$ and $(c,d)$ be two points on the curve of a continuous function such that $a>b $ and $c<d$ ...
0
votes
1answer
60 views

Evaluate h(x) = x + 1/x

Given $$ h(x) = x + 1/x $$ Find $$ h(a-1) $$ My answer is $0$, since when I put them in fraction form everything cancelled out.
2
votes
0answers
22 views

An identity involving Chain Function

A function $f \colon \mathbb Z^2 \to \mathbb R$ is called a chain function if all of the following conditions hold: $\forall x \in \mathbb Z \quad f(x,x)=0$. $f(1,0)=1$. $\forall x, \, y \in ...
0
votes
3answers
60 views

Find how far runners travel on a circular track (trig)

-How far has each runner traveled after 8 seconds? Though I just had to convert the rad/sec to rev/sec to get yards then multiply that by 8 seconds, but that isnt correct. Find the angle θ, in ...
0
votes
3answers
61 views

How to make a cos function into a sin function

I need to convert this equation into a sin function: f(x) = 12 cos(2x + 1) − 3 I know cos(x)= sin (pi/2 -x) but other than that I dont know how to solve this problem
1
vote
3answers
52 views

Find a function that is surjective and not injective

Rules: No piecewise functions. The function must be even, odd, or both even and odd. It cannot be neither. If this is impossible, prove why. This is just something I came up with for fun while ...
3
votes
6answers
157 views

How do I verify that $\sin (\theta)$ and $\cos (\theta)$ are functions?

I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there ...
1
vote
1answer
39 views

Algebra question: Do composite functions remain odd/even

“If $g$ is an even function and $h$ = $𝑓 ∘𝑔$, then $h$ is also an even function.” So tried it with one even function, $g(x)=x^2-5$ and an odd function and $f(x)=5x$ $$f(g(x))=f(g(-x))=25x^2-5$$ ...
1
vote
2answers
28 views

Analyzing a function's domain

Given the function $$f(x)=\frac{1}{x-8}$$ and the question: What is the domain of $f(x)$ I would normally look for values which $f(x)$ can not take. So I would check for plus infinity...Well $f(x)$ ...
0
votes
1answer
24 views

Problem with definition of amplitude of function

If $a$ and $b$ are respectively the minimum and maximum values of function $h$ then the amplitude of $h$ is: $$\frac{b-a}{2}$$ Why is this true?
3
votes
2answers
205 views

Finding Domain of a Function with a Fraction Inside a Square Root

I need to find the domain of this function: $$ f(x) = \sqrt{\frac{x-3}{x^2-3x+2}} $$ I understand that initially the denominator cannot be zero, and since it's just a formatted equation with roots 1 ...
2
votes
3answers
125 views

If $f(x) = 3x-4$ (functions, highschool)

if $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. I would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.
0
votes
2answers
36 views

Finding the domain of $f(x)=\ln(3x-4).$

I am trying to find the domain of $f(x)=\ln(3x-4)$. I cannot find out how to get the domain. but I did manage to get the vertical asymptote which is $x=4/3$.
1
vote
1answer
61 views

A simple function equation in calculus-1 course

Here is a homework question: $f^2(\ln x)-2xf(\ln x)+x^2\ln x=0,\ f(0)=0,\ f(x)=$? I don't know how to solve it. Thanks!
3
votes
1answer
124 views

Find a non constant function that is a quotient of two polynomial, for which: $f\left(x+\frac{1}{f(x)}\right)=f(x)f(-x)$

Before I post the problem, I want to ask what is wrong with the exactly same problem I posted three days ago, 'cause nobody seemed willing to answer it. The non constant function must satisfy the ...
1
vote
1answer
22 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.