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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3
votes
1answer
55 views

How do I solve $x=\log^e{(x+1)}$ analytically?

How do I solve the following, analytically? $$x=\log^e{(x+1)}$$ It looks like it should be simple, but whether I take the $e$th root of each side or take the $\log$ of each side (ending up with a ...
-3
votes
2answers
34 views

How would I be able to tell if some vector is in the span of a set of vectors?

Given the following, how would I be able to tell if b and c are in the span of the set of vectors S? Any help is appreciated. enter link description here
1
vote
1answer
27 views

A simple problem of the equation of a plane.

Two planes given $$x-y+z=5 , \hspace{0.5cm}x+y+z=3 $$ Their intersection is a line $l$.Find the equation of a plane such that the line $l$ is perpendicular to that required plane and this plane ...
0
votes
0answers
27 views

How to show that if average of squares equals square of average then all $X_i$ are equal?

Defining $A(X) = \sum_{i=1}^np_iX_i$, how would I show that given $A(X^2) = A(X)^2$, all $X_i$ must be equal? I tried by contrapositive - assume they are not equal and show that $A(X^2) \ne A(X)^2$. ...
3
votes
5answers
83 views

Show that $\gcd(a,b)>1$

Given are three natural numbers $a$, $b$ and $c$, for which $$\frac1a+\frac1b=\frac1c$$ Show that $\gcd(a,b)>1$. Could you someone provide a hint? I already tried algebraic manipulation, but I ...
-2
votes
1answer
28 views

Problem related to averages. [on hold]

The average height of the $10$ girls in class is $105$ cm. The average height of the $15$ boys in class is $125$ cm. What is the average height of all students in the class?
1
vote
2answers
37 views

integer $n$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square

Prove that the no integer $n\;,$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square. My Try:: We can write $(n^6+3n^5-5n^4-15n^3+4n^2+12n+3) = (n^3+an^2+bn+c)^2$ Now Here we have to ...
0
votes
1answer
14 views

How do I solve for x and y: x + 0.0467 + y = 1.000?

I am trying to find the isotopes for percent abundance question. I am looking over and answer and can't figure it out because of the math. Here it goes. 1) Set up a system of two equations in two ...
0
votes
0answers
30 views

find minimum and justify it

After having found the derivative of which if i am not mistaken is : I need to find the minimum of the function for which if I am not mistaken I equal to 0 the first derivative is this the right ...
0
votes
1answer
24 views

Value of $x$ where the graph lies below the graph of f(X)

From this question (How can I find the values of $x$ where a function lies below or above the axis?) I learn that "lies below" means $f(x)<0$ , now my question is , how can I check values that lie ...
-2
votes
1answer
58 views

How do I solve the inequality $25^{\sqrt x} < 5 \cdot 5^{\sqrt[3]{x}}$? [on hold]

Please help me solve this: $$25^{\sqrt{x}} < 5\dot \ 5^{\sqrt[3]{x}}$$
0
votes
1answer
17 views

How can I find the values of $x$ where a function lies below or above the axis?

Let's imagine this problem: Find the values of $x$ where the graph of $$f(x)= \frac{3x^2}{x^2-1}$$ lies below the $x$-axis. I know how to find the intercept $(0,0)$, but I don't understand what ...
-2
votes
0answers
23 views

Find the original quantity of water in the flask [on hold]

$10\%$ of salty sea water contained in a flask was poured out into a beaker. After this, a part of the water contained in the beaker was vapourised by heating and due to this the percentage of salt ...
0
votes
3answers
30 views

no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j, $ is

If $A = \left\{1,2,3,4\right\}$ and $B = \left\{1,2,3,4,5\right\},$ Then $(a)\; :: $ Total no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j, $ is $(b)\;\;::$ ...
0
votes
2answers
55 views

Equation $ \sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}=\sqrt[n]{b_1}+\sqrt[n]{b_2}+\cdots+\sqrt[n]{b_l} $

Let $k,l$ be natural numbers and $\{ a_i, b_i \}$ be real positive numbers such that $a_1\leq a_2 \ldots \leq a_k$, $b_1\leq b_2 \ldots \leq b_l$ and $$ ...
1
vote
4answers
43 views

Definite integrals involving $\ln x$

Alright, I have been working on this definite integral for the past couple days now and I can't for the life of me obtain the correct answer. I am not too sure where I am going wrong but I think the ...
0
votes
0answers
30 views

Logical problem about a guy's and 2 girls' motion on a line.

KS, Medy and Edi went to a party at Bau’s house. After it, they depart to their respective houses. Edi walks home, with a constant speed a, while KS and Medy rides a bike and drives home respectively ...
2
votes
2answers
29 views

Logarithmic function with strange bases

Given $\log_{4n} 40\sqrt{3} = \log_{3n} 45$, find $n$. I have rewritten $\log_{3n} 45$ as $\dfrac{\log_{4n}45}{\log_{4n}3n}$ and multiplied to get $\log_{4n} 40\sqrt{3}\cdot\log_{4n}3n = \log_{4n} ...
0
votes
1answer
15 views

Simplify complex fraction

I am working to simplify the equation 1+(1/x) divided by 1-(1/x), but i didnt get it. The solution given was to multiply by x/x and the answer is 1+(2/x-1) My solution was: (a) 1+(1/x) = (x+1)/x (b) ...
2
votes
1answer
20 views

Total Time Taken Question

Distance of chord = Time taken to "swim" to the desalination plant = I'm stuck here! The textbook working out is as such: I don't understand how they have the 'k' or 1/2 the runs river at ...
0
votes
1answer
23 views

Shortest Chord from origin to function

worked solution: Is this found using the distance of a line equation, where instead of co-ordinate points they use functions, so the two functions are g(x) and x (because the origin is on the ...
1
vote
1answer
86 views

Arithmetic sequence of natural numbers

Consider an arithmetic progression of natural numbers with a non-zero common difference. For each of the members of the progression its square root is taken, and if the square root is not an integer, ...
0
votes
2answers
36 views

Finding all the roots (rational, irrational, and complex) of a polynomial

$$x^6-64$$ I have already tried using synthetic division but get stuck after the 3rd round of division. Then I tried looking at it as a difference of squares. That didn't clear anything up either. ...
0
votes
1answer
31 views

Solve for $x$ an inequality with logarithms [on hold]

I am trying to solve this equation $\frac{(n(n-1))}{2} + X (2\log n +2) < nX $ I would like to solve it for X ? What should i do ? Thanks
5
votes
1answer
55 views

Algebra problem solve for a,b,c and d?

Can anyone find the values of these integers: a,b,c and d? $$1+\sqrt{2}+\sqrt{3}+\sqrt{6} = \sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$$ a+b+c+d = ? Thank you.
1
vote
2answers
61 views

Is my outcome wrong? (Evaluating a logarithm)

$$log\sqrt [ 4 ]{ x^2+y^2 } $$ $$log\sqrt { x+y } $$ $$logx^{ 1/2 }+log^{ 1/2 }$$ $$\frac { 1 }{ 2 }log (x+y)$$ The answer key saids: $$\frac { 1 }{ 4 } log(x^2+y^2)$$
1
vote
3answers
28 views

Not understanding what is going on in this problem (evaluating a logarithm)

$$\log({ \log }_{ 10 }10000)$$ Steps I took to solve this: ${ \log }_{ 10 }10000=4$ ${ \log }_{ 10 }4=y$ $10^{ y }=4$ ${ \log }10^{ y }=\log 4$ $y=\frac { \log 4 }{ \log 10 } $ doesn't seem to ...
0
votes
3answers
48 views

Write the equation of the tangent line of a circle

I'm totally lost with this question. I appreciate any kind of help. if the equation of a circle is $(x-3)^2+y^2=9$ Find : -Equation of the tangent line at $(2,2\sqrt2)$ -Equation of the tangent ...
1
vote
2answers
28 views

Stuck with finding the domain of a function with a logarithm

Find the domain of the function $$g(x)=\log_3(x^2-1)$$ This is what I got so far: $$\{ x\mid x^2-1>0\} =$$ $$\{ x\mid x^2>1\} =$$ $$\{ x\mid x>\sqrt { 1 } \}= $$ I don't know where to ...
0
votes
4answers
37 views

summation algebra for $\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1}$

Why does $\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1} = 1 + 2\sum_{n=1}^\infty x^n$? Shouldn't this be $1 + x + 2\sum_{n=1}^\infty x^n$ because of the $n+1$ in the second summation?
0
votes
0answers
54 views

Solving Systems of equations for $(x,y)\in\mathbb {R}^2$

So I'm working on solving a couple of system of equations: $$ \text{Let} \ a,b \ \text {be a positive real number with} \ a\neq b \ \text{Solve the system:}$$ ...
0
votes
2answers
48 views

Express $\frac{1}{(3-\sqrt{2})^2}$ in the form $p+q√2$ [on hold]

Both $p$ and $q$ have to be rational numbers. Anyone have a step by step solution? I have tried to expand the bracket in the denominator and then multiplied top and bottom by the conjugate but I ...
2
votes
1answer
57 views

Finding the integer solutions of the equation $3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14$

$ 3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14 $ . I already solved this using the Cauchy–Schwarz inequality and got $x=4$ and $y=5$. But I'm sure there is a prettier, simpler solution ...
-1
votes
1answer
33 views

What is the Algebra involved in finding the domain for $\sqrt x\le 2$ [on hold]

Would like to know how one would solve this algebraically. Show all steps and keep in mind that I am a pre-calculus student.
2
votes
3answers
91 views

Problem with roots

I am having a few problems with roots. This is apart of a larger question where I am taking the derivative of of a function. I know I got the first part right (answer key) but when I plug in root 2 ...
1
vote
4answers
61 views

Rationalise $\frac{2}{\sqrt{12}}$ fully

I keep coming up with $\frac{\sqrt{6}}{6}$ but I don't think that it's right. Can you divide a surd by a common factor like $2$ to get rid of the denominator? Would really appreciate it if someone ...
0
votes
2answers
17 views

Simplify this expression 1/(3-√2)^2

How to simplify 1/(3-√2)^2 ? Does the ^2 mean you do something different? I know that you need to rationalise he denominator by multiplying top and bottom by 3+√2 but I don't know what happens with ...
0
votes
3answers
42 views

Simplify in the form $p+q\sqrt2$

How to simplify fully as far as possible: $$\frac{\sqrt 2}{3\sqrt2-4}$$ Can anyone explain to me how to know when a surd expression is completely simplified?
0
votes
2answers
33 views

Help with this surds question please!!

It is given that $$k=\frac{\sqrt 3-\sqrt 2}{\sqrt 3+\sqrt 2}$$ Express $k$ in the form $p+q\sqrt 6$, where $p$ and $q$ are integers. Express $1/k$ in the form $r+s\sqrt 6$, where $r$ and $s$ are ...
1
vote
4answers
69 views

Example of a non-trivial function such that $f(2x)=f(x)$

Could you give an example of a non-constant function $f$ such that $$ f(x) = f(2x). $$ The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the ...
3
votes
3answers
129 views

Trigonometry Difficult Question [on hold]

If $\cos x + \cos y = a$ and $\sin x + \sin y = b$. Find $\cos(x+y)$ and $\sin(x+y)$. I only need some hints to start as I am not able to get any way to go forward to.
2
votes
2answers
42 views

Differentiability and continuity of a trig function

Here's a problem I'm having a lot of trouble with: We have the following function: $f(t) = t^2\cos(\dfrac{1}{t})$ for $t \neq 0, f(t) = 0$ for $t = 0$. Show $f$ is continuous ...
0
votes
1answer
18 views

If $x$,$y$ $\in[2,\infty)$, then $xy - 2x - 2y + 6$ $\in$ $[$$2$,$\infty$$)$ [on hold]

IF $x$,$y$ $\in$ $[$$2$,$\infty$$)$ , prove that $xy - 2x - 2y + 6$ $\in$ $[$$2$,$\infty$$)$
1
vote
2answers
24 views

Present Value (Interest)

The question goes like this: What deposit made today will provide for a payment of 1000 in 1 year and 2,000 in 3 years, if the effective rate of interest is 7.5%? The answer given by the book is ...
1
vote
2answers
77 views

Show that there exists no integer b such that f(b) is 1993.

We are given a polynomial $f$ with integer coefficients such that for 4 distinct integers $a_1,a_2,a_3$ and $ a_4$, $f(a_1)=f(a_2)=f(a_3)=f(a_4)=1991$. Show that there exists no integer $b$ such that ...
0
votes
1answer
32 views

How to find the common base of terms in an expression?

I'm teaching myself basic algebra from a book and am stuck on a question. In the current section it is about expressing numbers as powers of the same base. So $9$ maybe expressed as $3^2$. Another ...
0
votes
1answer
27 views

Definition of the Coefficients of a Quadratic Polynomial

Hey guys I have a pretty straight forward question that I was wondering about. Would $c$ in the following equation be considered a coefficient and constant or just a constant? $f(x)= ax^2+bx+c$. ...
0
votes
5answers
53 views

Find domain $\;g(x)= \sqrt{x^2-9}\;$

Okay $g(x)= \sqrt{x^2-9}$ thus, $x^2 -9 \ge 0$ equals $x \ge +3$ and $x \ge -3$ thus the domains should be $[3,+\infty) \cup [-3,\infty)$ how come the answer key in my book is stating ...
0
votes
0answers
38 views

Relationship for $\log(A_1+A_2+\cdots+\cdots+A_n)$

It's a very well know fact that $$ \log\left(\prod_{i=1}^n A_i\right)=\sum^{n}_{i=1}\log(A_i) $$ Can we say anything about $$ \log\left(\sum_{i=1}^n A_i\right)=\text{ ????} $$ My question is ...
0
votes
0answers
20 views

Restrictions to domain and range

Amelia is planning the trajectory of her next flight using the altitude and distance from Paris, France. She has determined her function to be $\displaystyle f(x) = -4x + 102$. Based on the situation ...