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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4
votes
0answers
30 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
57 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
76 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 ...
-2
votes
1answer
29 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. ...
0
votes
1answer
62 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
29 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
30 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 ...
25
votes
6answers
1k 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
27 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
50 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
15 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
49 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
5answers
105 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
34 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
35 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
27 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 ...
1
vote
3answers
40 views

Multiplicity of a root of a polynomial

:) It's true that, if a polynomial has a root (let's say, k, for example) with multiplicity n (n>1, for n integer), then it's true that the derivate polynomial have k as a root with multiplicity ...
5
votes
2answers
35 views

Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
4
votes
1answer
64 views

If $x=(9+4\sqrt{5})^{48}=[x]+f$ . Find $x(1-f)$.

If $x=(9+4\sqrt{5})^{48}=[x]+f$, where $[x]$ is defined as integral part of $x$ and $f$ is a fraction, then $x(1-f)$ equals . $\color{green}{a.)\ 1} \\ b.)\ \text{less than}\ 1 \\ c.)\ ...
-4
votes
3answers
70 views

The limit of iterated square root with multiplication under the root, $\sqrt{ a \sqrt{ a \sqrt{a \cdots}}}$ [duplicate]

$$ \sqrt{ a \sqrt{ a \sqrt{a \cdots}}}=\text{ ?} $$ options were given as $0$ $-a$ $a$ $1$ i did not know how to solve it or what it was related to. Could anyone please explain the concept ...
-3
votes
1answer
30 views

Coding Matrix Problem

(A) Use the coding matrix $A=\left[\begin{smallmatrix}1&3\\5&-3\end{smallmatrix}\right]$ to encode the word jump (B) Using it's Inverse $A^{-1}= ...
0
votes
2answers
28 views

Why does $\sqrt{1 + 4x^2} > 2|x| \implies -1 - \sqrt{1 + 4x^2} < -2x$?

Given $f(x) = \frac{-1 - \sqrt{1 + 4x^2}}{2x}$, if $x > 0$, show that $f(x) < -1$. Solution: Note that $\sqrt{1 + 4x^2} > 2|x|$. So if $x > 0$, then $-1 - \sqrt{1 + 4x^2} < ...
-3
votes
2answers
30 views

want to know about probability? [on hold]

A certain fruit stand sold apples for \$0.70 each and bananas for \$0.50 each. if a customer purchased both apples and bananas from the stand for a total of \$6.30, what total number of apples and ...
-1
votes
2answers
37 views

What is a proof for the product rule for logarithms? [on hold]

What are some proofs for this? I'd like to see more than one type if possible.
-1
votes
6answers
119 views

What is the mathematical symbol for the sum of numbers

For example, when $n=5$, what is the symbol for $5+4+3+2+1$?
0
votes
1answer
41 views

summation problem [on hold]

what is the result for the following double summation: $\sum\limits_{i \neq j}^{\infty}\alpha^i\alpha^j$ where $ i, j =0,1,2,.......$
0
votes
2answers
37 views

From the graph find the number of solutions.

The figure below shows the function $f(x)$ . How many solutions does the equation $f(f(x))=15$ have ? $a.)\ 5 \\ b.)\ 6 \\ c.)\ 7 \\ d.)\ 8 \\ \color{green}{e.) \ \text{cannot be determined from ...
0
votes
2answers
75 views

Make $\frac{a}{b+c} = \frac{a}{b} + \frac{a}{c}$ right

I have this math exercise where I have to make the equation $$\frac{a}{b + c} = \frac{a}{b} + \frac{a}{c}$$ right. Can anyone help me please? I have already tried things like $$a\left(\frac{1}{b} + ...
1
vote
1answer
80 views

Need hint to solve a nasty integral.

Let $f(x)=\frac{x+2}{2x+3}$, $x>0$. If $$\int \left( \frac{f(x)}{x^2} \right)^{1/2}dx=\frac{1}{\sqrt{2}}g \left(\frac{1+\sqrt{2f(x)}}{ 1-\sqrt{2f(x)}} \right) -\sqrt{\frac{2}{3}}h ...
5
votes
2answers
91 views

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer. Doesn't this problem seem a little out of the box? It seems beautiful, but I don't have an idea on how to start. Calculating the value does ...
1
vote
1answer
55 views

Prove $e^x$ limit definition from limit definition of $e$.

Is there an elementary way of proving $$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n,$$ given $$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n,$$ without using L"Hopital's rule, Binomial Theorem, ...
1
vote
2answers
41 views

Concluding three statements regarding $3$ real numbers.

$\{a,b,c\}\in \mathbb{R},\ a<b<c,\ a+b+c=6 ,\ ab+bc+ac=9$ Conclusion $I.)\ 1<b<3$ Conclusion $II.)\ 2<a<3$ Conclusion $III.)\ 0<c<1$ Options By ...
6
votes
2answers
72 views

Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$.

I need to prove, that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. Prove that $$x^m+x^{-m}=P_m (x+x^{-1} )=a_m (x+x^{-1} )^m+a_{m-1} (x+x^{-1} )^{m-1}+...+a_1 (x+x^{-1} )+a_0$$ on ...
-2
votes
5answers
97 views

What is the value of $x*y$? [on hold]

Given that $$\left(\frac{x}{y}\right)^{-2} + \left(\frac{y}{x}\right)^{-2} = \frac{10}{3}$$ find the value of $x*y$. My question is, can we calculate the value of $x*y$ or not? If yes, then ...
4
votes
0answers
59 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
1
vote
1answer
63 views

Why memorize trig identities?

I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my ...
0
votes
3answers
44 views

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$?

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$? I am a high school student and stuck at this simple question on domains and ranges of functions. I have done the ...
0
votes
1answer
41 views

Finding value of an exponential equation [duplicate]

If $ x^2=3x-1$ then find the value of $(x^6+1)/x^3$ I used the quadratic but it became too complicated
1
vote
5answers
150 views

Finding value of an expression

If $x^2-3x-1=0$ then find the value of $(x^6+1)/x^3$ I tried to solve the quadratic but it became too complicated any way of doing this without a calculator
1
vote
4answers
258 views

Negative exponents and positive numbers.

Why is it that when we raise a positive number to a negative power, we don't get a negative number?
4
votes
2answers
73 views

If $\frac{(b−c)}{a} + \frac{(a+c)}{b} + \frac{(a−b)}{c}=1$ and $a-b+c \neq 0 $, then prove that $\frac 1a = \frac 1b + \frac 1c$

The question given is If $\dfrac{(b−c)}{a} + \dfrac{(a+c)}{b} + \dfrac{(a−b)}{c}=1$ and $a-b+c \neq 0 $ then prove that $\dfrac 1a = \dfrac 1b + \dfrac 1c$ I tried to take $abc$ on the right ...
1
vote
0answers
50 views

A geometric interpretation of a geometric series

The following code is compiled by TikZ to draw a right triangle so that the enclosed area is partitioned by infinitely many triangles similar to itself. I saw on ...
-1
votes
3answers
33 views

Question in fraction [on hold]

I find it difficult to solve this. Please help me out.... At a party, the number of men was $\frac{2}{5}$ the number of women. After $144$ women left, the number of women become $1\frac{1}{2}$ ...
0
votes
5answers
46 views

Help Solving a Simultaneous Equation.

Im currently doing my Kumon (A math tutoring center I guess) homework, and Im having a bit of difficulty answering a simultaneous equation, involving $x$ and $y$ variables to the second power. School ...
0
votes
1answer
41 views

The value of $x$ for which function attains max value

At what value of $x,\ x\in \mathbb{Z}$ will the function $\dfrac{x^2+3x+1}{x^2-3x+1}$ attain its maximum value . $\color{green}{a.)\ 3 }\\ b.)\ 4 \\ c.) -3 \\ d.)\ \text{none of these} \\ $ ...
5
votes
1answer
63 views

Inequality with reciprocals of $n$-variable sums

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Is it always true that $$\sum_{i=1}^n\frac{1}{a_i}-\sum_{1\leq i<j\leq n}\frac{1}{a_i+a_j}+\sum_{1\leq i<j<k\leq ...
2
votes
3answers
41 views

Reasoning about numbers close to two other numbers $a,b$ (inequalities)

Let $a < b$ and $0 < \varepsilon < (b - a)$ and let $x, y \in \mathbb R$ be such that $$ | x - a | < \frac{(b - a) - \varepsilon}{2}, \qquad | y - b | < \frac{(b - a) - ...
1
vote
4answers
116 views

Why is $\sqrt{x^2}= |x|$ rather than $\pm x$? [duplicate]

Shouldn't the square root of a number have both a negative and positive root? According to Barron's, $\displaystyle \sqrt{x^2} = |x|$. I don't understand how.
1
vote
2answers
47 views

Find $f\circ f$ if $f(t)=\dfrac{t}{(1+t^2)^{1/2}},\ \ t\in \mathbb{R}$

Find $f\circ f$ if $f(t)=\dfrac{t}{(1+t^2)^{1/2}},\ \ t\in \mathbb{R}$ $ a.)\ \dfrac{1}{(1+2t^2)^{1/2}} \\ \color{green}{ b.)\ \dfrac{t}{(1+2t^2)^{1/2}}}\\~\\ c.)\ (1+2t^2)\\~\\ d.)\ ...