Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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1
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3answers
30 views

Finding X from Exponential Equations

$$2^x \cdot 4^{1-x}= 8^{-x}$$ I wrote all the base numbers as a power of 2 but I'm not sure what to do after.
-1
votes
2answers
28 views

Express $x+y+z$ in terms of $a$ and $b$ [on hold]

If $A = X + Y$ and $B = X + Z$, find the value of $X+Y+Z$ in terms of $A$ and $B$.
-3
votes
2answers
57 views

Solving for $x$ in $A=B\cdot \cos(x)+C\cdot \sin(x)$ [duplicate]

I´m working on a little paper, and I want to know if it´s possible in any way to solve this: $$A=B\cdot \cos(x)+C\cdot \sin(x)$$ $A$, $B$ and $C$ are known. I need a way to get the $x$ without using ...
-4
votes
1answer
25 views

Evaluate $\log 64$ using the change of base formula? [on hold]

Is that even possible? I mean, there is no base.
0
votes
1answer
26 views

What is the proof for this sum of sum generalized harmonic number?

I believe this sum: $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}$$ to be equal to $$\frac 12((H_k^{s})^2-H_k^{2s})$$ where $H_k^{s}$ is the generalized harmonic number. I only discovered this by ...
-1
votes
2answers
58 views

In the equation $2n^3 + 3n^2 = 500,000$, what does $n$ equal? [on hold]

In the equation $2n^3 + 3n^2 = 500,000$, what does $n$ equal?
1
vote
1answer
99 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
3
votes
6answers
72 views

How do you show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist? [on hold]

How can I show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist ?
0
votes
0answers
30 views

Determine the minimum and maximum angles, to the nearest tenth of a degree, that a pipe can make with the horizontal.

For residential drains, a horizontal pipe needs to have a minimum slope of 1/4 inch per foot and a maximum slope of 1/2 inch per foot for waste to drain properly. This means that for every horizontal ...
1
vote
4answers
41 views

Trigonometry equation. Not sure about solution.

The equation goes as follows: $$\sin x +\cos x = 1 + \sin x \cos x$$ and here is how I solved it: $$(\sin x+\cos x)^2=(1+\sin x\cos x)^2$$ $$\sin^2x+2\sin x\cos x+\cos^2x=1+2\sin x\cos ...
1
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2answers
51 views

solve $\sqrt{x+7}<x$ for $x\in \mathbb{R}$

solve $\sqrt{x+7}<x$ I tried $\sqrt{x+7}<x\\ x+7<x^2\\ x^2-x-7>0\\ x\in \left(-\infty, \dfrac{1-\sqrt{29}}{2}\right) \cup \left( \dfrac{1+\sqrt{29}}{2},+\infty\right) $ I m not ...
-5
votes
3answers
58 views

What's the value of $i^i$? [duplicate]

What's the value of $i^i$?Is it real or imaginary?[$i$ here denotes imaginary number.]
-1
votes
1answer
29 views

Inverse of rational function

I need help with this question: Determine whether the given function is one-to-one, and if so, find the inverse: $$ f(x) = 5x + \frac{2}{x} $$ Wolfram says the answer is $\frac{1}{10}\left(x ...
3
votes
3answers
36 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
0
votes
0answers
14 views

dependent variable change attribution

I have a model that is generally represented as $y = w x z$. Period over period, each of these independent variables will change, and therefore so will the dependent variable. I am trying to ...
-2
votes
1answer
19 views

What are the coordinates of a point given its distance from another point?

If the abscissa of a point is twice the value of the ordinate and has a distance of $2\sqrt{17}$ units from the point $(4,-5)$, what are the coordinates of the point?
0
votes
1answer
26 views

Write all elements of A.A = {$x|x^2<x<10$,x is a whole number}. Answer: A ={$x|x^2+1=0$}.Explain like i'm five.

Write all elements of A.A = {$x|x^2<x<10$,x is a whole number}. Given Answer: A ={$x|x^2+1=0$}. Is this a typo?
4
votes
3answers
34 views

finding $a_1$ in an arithmetic progression

Given an arithmetic progression such that: $$a_{n+1}=\frac{9n^2-21n+10}{a_n}$$ How can I find the value of $a_1$? I tried using $a_{n+1}=a_1+nd$ but I think it's a loop.. Thanks.
1
vote
3answers
144 views

What are the products of real solutions of this equation?

How can I solve $\:\: \log^2_{1/2}(4x)+\log_2\hspace{-0.06 in}\left(\hspace{-0.06 in}\frac{x^2}{8}\hspace{-0.06 in}\right)=8 \;$ ? I have tried the elementary for logarithms simplifying the terms in ...
1
vote
3answers
35 views

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$ The answer should be: $y = \frac{1}{12} x^2 -3$ But how to arrive at the answer? I tried replacing r with $\sqrt{x^2 + y^2}$, then ...
0
votes
2answers
52 views

Explain the role of the numerator and denominator of a rational exponent such as $\left(\frac{27x^3}{8y^9}\right)^{-\frac{5}{3}}$ [on hold]

So I understand how to solve this problem, (8y^9 / 27x^3)^5/3 ((2y^3)^5 / (3x)^5 (2y^3)^5 / (3x)^5 (32y^15) / (243x^5) but i am confused as to what the direct role of the numerator and denominator is ...
2
votes
3answers
55 views

Least Common Denominator: $ \frac{\sqrt{x}}{x}+\frac{\ln\ x}{2\sqrt{x}} $

$$ \frac{\sqrt{x}}{x}+\frac{\ln \ x}{2\sqrt{x}} $$ I have tried combining these two fractions; however, I keep getting stuck. $$\frac{2\sqrt{x}}{2\sqrt{x}}\cdot\frac{\sqrt{x}}{x}+\frac{\ln\ ...
0
votes
0answers
35 views

What is the difference between the scalar and vector components of a vector?

What is a scalar component of a vector and what is a vector component of a vector. suppose a vector is making and angle theta with the origin then in my book it is written that its x component is the ...
1
vote
2answers
79 views

Proving inequalities using Calculus

In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example $$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$ How would you use ...
3
votes
0answers
40 views

Solving for a variable in an inverse function

I was asked to solve this formula for $R_2$: $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$ So I did the following: \begin{align*} \frac{1}{R_2} &= \frac{1}{R} - ...
0
votes
3answers
60 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
4
votes
2answers
33 views

Solve for $x$ in: $e^{2\ln(x)-\ln(x^2+x-3)} = 1$

So the question is to solve for x in: $$e^{[2\ln(x)-\ln(x^2+x-3)]} = 1$$ I took the natural log of both sides, and simplified. Here is what I've gotten it down to: $$2\ln(x) = \ln(x^2+x-3)$$ And I'm ...
1
vote
1answer
19 views

Arithmetic, Geometric and Harmonic prove equation.

If $a, b, c, d$ be in Arithmetic Progression, $a, e, f, d$ be in Geometric Progression, and $a, g, h, d$ in Harmonic Progression respectively; prove that $ad=ef=bh=cg$.
3
votes
1answer
34 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
4
votes
2answers
62 views

How do i find $\tan(\theta)$ such that : $\frac{16}{\sin^6(\theta)} + \frac{81}{\cos^6(\theta)}=625$??

How do i find $\tan(\theta)$ such that :$$\frac{16}{\sin^6(\theta)} + \frac{81}{\cos^6(\theta)}=625$$? Note : i used some trigono-form but sorry i didn't succed . Thank you for any help.
0
votes
1answer
32 views

Functions of modulus

How do I calculate the range of any modulus function? I know that if $x <2$ then it's expansion is negative and if $x>2$, it's expansion is negative, but will it help? Consider an example, $$f ...
0
votes
4answers
50 views

Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
0
votes
3answers
41 views

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$ The answer in the book is ln $\frac {\sqrt{x}}{x+1}$ If am not allowed to to cancel terms ...
5
votes
1answer
59 views

Function equation, find the function evaluated at the certain point.

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ The constant term, $a_0 = f(0) = 1$. Let: ...
4
votes
4answers
81 views

$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$?

The question given is Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$. What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the ...
3
votes
8answers
96 views

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$ I don't know how to find all possible values. I tried making the lower of the two roots as ...
-3
votes
1answer
37 views

Use the graph of Y=f(x) shown below to answer the following questions

https://www.flickr.com/photos/134404416@N03/shares/6p9h7f I tried answering some of the questions as you can see in the picture link provided, but I just am not sure if my answers are even right. ...
1
vote
2answers
78 views

How to show this fraction is not an integer

Suppose $k\geq 2$ is an integer. I want to show $$\frac{1+k+k(k-2)}{1+\frac{k-1}{k}+\frac{(-1-\sqrt{k-1} )^2}{k(k-2)}}$$ is not an integer. It is equal to $$\frac{(k-2) k (k^2-k+1)}{2 (k^2-2 ...
0
votes
3answers
31 views

How do you determine the end behavior of a rational function?

Example $$\frac{6x + 2}{x^2 - 9} = \frac{6x + 2}{(x + 3)(x - 3)}$$ I know how to find the vertical and horizontal asypmtotes and everything, I just don't know how to find end behavior for a ...
0
votes
2answers
36 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let ...
32
votes
6answers
2k views

Is there something between summation and integration?

Let's take a general function $f(x)$, we can do a summation like: $$\sum_{k=m}^n f(k)$$ And we can do an integration like: $$\int_a^bf(k)dk$$ The basic difference between the two operation is that ...
-2
votes
3answers
29 views

Practice math question gkt [on hold]

How many 3/8 pound hamburger patties can be made from 4 1/2 pounds of ground beef?
2
votes
3answers
52 views

Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$ x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 $$ given that there are no multiple roots. I am in a weird situation here. ...
1
vote
1answer
16 views

Recursive Equation : $X_t=-\sum_{j=1}^{\infty}\phi^{-j}W_{t+j}$

$X_t=\phi X_{t-1}+W_t$ $\Rightarrow X_{t-1}=\frac{1}{\phi} X_{t}-\frac{1}{\phi}W_t$ Where , $|\phi|>1$ . But how does the following recursion relation occur : ...
1
vote
2answers
53 views

What is the product of results of this equation?

How to get product of results by $x$? $$(25+x)^{1/3} + (3-x)^{1/3} = 4$$ I have tried to to get both sides on cube but I got nothing.
1
vote
6answers
131 views

If $a+b+c+d=1$ then why is the maximum value of $(a+1)(b+1)(c+1)(d+1)$ is ${\left(\frac{5}{4}\right)}^4$?

What I know is that for equations of type $x+y=8$, $xy$ attains its maximum value when $x=y$ and this can be proved by either solving the quadratic equation with completing the squares or finding the ...
3
votes
2answers
36 views

Prove by induction that for the Fibonacci numbers $F(n)$ with $n \ge 6$, $F(n) \ge 2^{n/2}$

Prove by induction that $F(n) \ge 2^{n/2}$ for $n \ge 6$ I've done the following steps: 1) Base case: $F(6) = 8$, $2^{0.5 \cdot 6} = 8$, base case proved. 2) Induction: let's assume that $F(k) ...
1
vote
2answers
26 views

Quadratic Absolute Value Inequality

Problem: Find all $x$ such that $|x^2-3x+1|<1$ I can't understand how to get started with this. I've never tried to solve quadratic Inequalities before. At first I thought of working with the ...
-1
votes
1answer
41 views

Find all possible values of $\phi$: $2(2^{\phi}-1)\phi^2 + (2^{\phi^2}-2)\phi = 2^{\phi+1}-2$ [on hold]

Find all possible values of $\phi$ in the following expression: $$2(2^{\phi}-1)\phi^2 + (2^{\phi^2}-2)\phi = 2^{\phi+1}-2$$
0
votes
2answers
29 views

How do I reverse the smooth-step equation?

I'm using the "smooth step" equation for an easing curve: $y = 3x^2 - 2x^3$ I would like to reverse this equation so that given y, I can find ...