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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0
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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
92 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
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3answers
44 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
71 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
132 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
19 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
79 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
54 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
39 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 ...
6
votes
2answers
109 views

Prove that $2^n +1$ in never a perfect cube

Prove that $2^n +1$ in never a perfect cube I've been thinking about this problem, but I don't know how to do it. I know that if $m^3=2^n+1$, then $m$ should be an odd number, but I 'm not able to ...
0
votes
2answers
50 views

How does one Graph $y= \sin (x/8)$?

I studied trigonometric functions this summer, however, I am lost as to how to apply what I learned to this problem. The question asks me to plot this function. 1) It is a sine function so the ...
0
votes
4answers
30 views

Evaluate coshX given that tanhX

Whilst working out some hyperbolic evaluation questions, I've come across this particular one. So far with any question I've come across I've simply tackled it step by step using hyperbolic ...
0
votes
1answer
11 views

Simple Growth rate question

Company X has 7% market share and is growing 8% per year. Company Y has 10% market share and is growing 2% per year. About how many years will it take X to pass Y? Show your work.
-1
votes
0answers
21 views

What is the vertex for the graph below? [closed]

What is the vertex for the graph below?
0
votes
1answer
13 views

Find an equation of a quartic function with 2 points and a tangent point

Find an equation of a quartic function if the curve passes through the points $(-3,0)$, $(-1,0)$ and is tangent to the $x$-axis at $(2,0)$
0
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0answers
14 views

Latitude and longitude of the destination of an airplane, given its speed and direction of travel

Problem A plane leaves an airport X, $20.6$ degree east and $36.8$ degree north, and flies due south along the same longitude for 8hrs at the rate of 1000km/h to another airport Y, $20.6$ degree east ...
0
votes
0answers
30 views

College Interest rate by yearly payments

Problem A loan of 20; 000, made at an interest rate of 6%, is to be repaid by level yearly payments for 10 years, beginning 1 year after the loan is advanced. Just before making the seventh ...
5
votes
1answer
49 views

Two sequences $a_{2n}=a_n+1, a_{2n+1}=a_{n}+2$ and $b_{3n}=b_n+1, b_{3n+1}=b_n+2, b_{3n+2}=b_n+3$

Let us consider two sequences $$a_{2n}=a_n+1, a_{2n+1}=a_{n}+2, a_1=1,a_2=2$$ and $$b_{3n}=b_n+1, b_{3n+1}=b_n+2, b_{3n+2}=b_n+3, b_1=1,b_2=2,b_3=2.$$ Prove that $a_{2^n} < b_{2^n}$ for ...
-2
votes
2answers
34 views

Which of the following is a situation in which an equation cannot be solved using the quadratic formula? [closed]

Which of the following is a situation in which an equation cannot be solved using the quadratic formula? A.) The coefficient of the $x^2$ term is zero B.) The right hand side of the equation is zero ...
-1
votes
1answer
11 views

comparing cake by who have better option for money [closed]

a bakery sell a "9 by 13" cake for the same price as an 8" diameter around cake .if the round cake is twice the height of the rectangular,which option gives the most cake for money
0
votes
1answer
35 views

Find the equivalence class containing the element

Consider the group G = {1, 3, 5, 7} under multiplication mod 8. Consider the subgroup H= {1,3}. Find the equivalence class containing the element 5 using the relation ~R. I am very stuck on this ...
0
votes
4answers
61 views

If statements in equations?

When creating an equation is there any way to let someone know that if this number doesn't fit into this then use this other equation? Ex: $x>1$ then use $x+y$, or if $x<1$ then use $x-y$. Or ...
1
vote
1answer
29 views

Exponential Functions Proof

I cannot find any material similar enough to this problem to be of use. the problem states, if $$ f(x)=5^x$$ show that $$\frac{f(x+h)-f(x)}{h} = 5^x\left(\frac{5^h -1}{h}\right)$$ so I begin to ...
-1
votes
3answers
154 views

Let $a$ be a real number, such that $a^7,\,a^{10}\in\mathbb Q$. [closed]

Let $a$ be a real number so $a^7,a^{10}\in$ $\mathbb Q$. Can we prove that $a\in\mathbb Q$. Could you please provide a hint?
-1
votes
1answer
19 views

Biking uphill and downhill

During an interview, I was asked "If you can bike 20 mph uphill and 30mph downhill, and you have 1 hour to bike, how far or how long should you ride uphill before turning back." While a very ...
3
votes
2answers
73 views

Tough trigonometric identity

Prove that $$\cot 13^o\cot 23^o \tan 31^o\tan35^o\tan41^o = \tan 75^o$$ I managed to rearrange it to the form $$\tan 31^o\tan 35^o\cot 49^o = \cot 15^o\tan 23^o\cot 77^o$$ and in this form we have ...
0
votes
1answer
26 views

Making a Piecewise Function into a single expression

A phone company gives a 25.00 dollar flat fee up to 200 minutes then .07 dollars for every minute afterwards. Build a function to find the price of any amount of minutes. Not in a piecewise function, ...
0
votes
0answers
101 views

Why is Wolfram Alpha wrong?

I calculated $$\tan 75^o - [\cos 13^o\cdot \cot 23^o \cdot \tan 31^o \cdot \tan 35^o\cdot \tan41^o]$$ and I got a nonzero answer: ...
0
votes
3answers
53 views

Real numbers determine rational numbers

Determine the rational numbers $x,y$ ,knowing that $x(1+\sqrt2)^2$ + $y(1-\sqrt2) = 1 $ My result is $3x$ + $y$ - $y\sqrt2$ = $1$ I'm not sure how to continue this.
1
vote
3answers
111 views

Solving $\log(x) = x-1$?

One can use Taylor series of the log or exp function to get the result that $x = 1$. I was wondering if there is any other simple solutions. Thanks a lot!
0
votes
3answers
22 views

Rearranging forumlas

How do I rearrange this formulae? I need to make $x$ the subject. $a)\ \ $ $k = \pi(x - t)$ I think it involves factorising but I'm not sure, I tried dividing the $(x - t)$ from both sides to get: ...
0
votes
2answers
48 views

Determine Whether the Function is even, odd, or neither $g(x) = 1-x^4$

These questions give me a problem since the rules of distribution seem not to apply to them, for example: Determine whether the following function is even, odd, or neither? $g(x)= 1-x^4 $ $g(-x) =1 ...
-4
votes
1answer
32 views

How much will tea cost if…

So, let's say I have tea leaves that cost me $\$7.40$ per kg and I'm ordering $160\text{ kg}$. All up the tea would cost me $\$1184$. Now, let's say I wanted to sell that tea in $30\text{ g}$ bags ...
1
vote
1answer
23 views

Difficult Polynomial Question: Finding the sum of specific coefficients of an expanded polynomial

When expanded, the product $(x+2)(x+3)(x+4)\cdots(x+9)(x+10)$ can be written as $a_9x^9+a_8x^8+...+a_1x+a_0$. What is the value of $a_1+a_3+a_5+a_7+a_9$?
0
votes
4answers
56 views

Proof that $n \geq (1+\frac{1}{2n})^n$

This is a part of a larger proof I am doing by induction for an exercise, but I've gotten stuck on this part: $$\forall n \in \mathbb{N}, n > 1 \implies n \geq (1+\frac{1}{2n})^n$$ Any pointers ...
1
vote
3answers
87 views

What exactly happens in the algebraic steps here?

$$ \frac{n(n+1)}{2} + (n+1) = (n+1)(\frac{n}{2} + 1) = \frac{(n+1)(n+2)}{2} $$ I don't understand what happens from the first to the second and from the second to the third one.