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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1answer
348 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
3
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1answer
28 views

From a ratio $q_1 = x_1/x_2$, how do I compute $q_2 = x_2 / (x_1 + x_2)$?

The question is mostly in the title. If I have a numerical value for a ratio $q_1 = x_1/x_2$, how do I compute $q_2 = x_2 / (x_1 + x_2)$? For example, if $x_1 = 1$ and $x_2 = 8$, we would be given ...
0
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2answers
266 views

What are the parameters of a parabola

In the following figure I understand the $bx+c$ part. It is simply the equation of a line. But I don't understand where did $ax^2$ came from? What exactly is it? What does $a$ tell us about a ...
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3answers
113 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 ...
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2answers
30 views

Trouble factoring polynomial

So I've gotten pretty far on this problem but I'm unable to complete it. I don't want to resort to the quadratic formula because this section of my review sheet specifically tells me not to use it. ...
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3answers
98 views

How come 0^0 = 1 [duplicate]

Consider this: 0^0 = 1 0^1 = 0 0^2 = 0 0^n = 0 (for int n > 0) So how come 0^0 = 1, how can you get something out of ...
1
vote
1answer
43 views

Trouble with factoring polynomial fractions and understanding wolfram alpha

I hope the community can excuse me if I'm making excessive posts, I have a calculus quiz tomorrow and I want to be as prepared as possible. I'm going through a review problem on wolfram alpha and I ...
3
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3answers
242 views

Question about factoring polynomial fraction

I ran into this problem in the review section of my math text but I'm not sure how to go about solving it. $$\frac{2x(x^2-9) - x^2(2x)}{(x^2-9)^2} = 0$$ I can't find a way to cancel the numerator ...
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2answers
474 views

software to solve system of nonlinear equations

I am looking for a software to solve system of nonlinear equations. It would be great if the software can satisfy the following requirements It can support symbolic computation. It deals well with ...
2
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1answer
39 views

How to simplify this expression that contains exponential terms?

In a multiple choice exam , I encountered the following question. The answer to the question is $$ \frac{17}{8}.$$ The question is: $$\frac{16^{x+1}+4^{2x}}{2^{x-3}8^{x+2}} \text{ is ? }$$
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2answers
78 views

Help in understanding this absolute value question

I would like to understand the answer to the following problem. Express the following with at least one less pair of absolute value signs: $$|(|a + b| - |a| - |b|)|$$ I know the answer is $|a| ...
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2answers
37 views

Point of interesection between ln and exponential functions

$\ln(x) = 1 - 1.2^x$ Looking at the functions graphically the solution is β‰ˆ 0.85. Is there a way to algebraically determine the precise point of intersection? I can isolate one "x" or the other, ...
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1answer
2k views

Help solve an equation

I'm preparing for the SAT and tripped over the following problem: $(x-8)(x-k) = x^2 - 5kx + m$ "In the equation above, k and m are constants. If the equation is true for all values of x, what is ...
3
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6answers
193 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 ...
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2answers
266 views

$a-b,a^2-b^2,a^3-b^3…$ are integers $\implies$ $a,b$ are integers?

Let $a,b$ be distinct real numbers such that $a^n -b^n$ is integer for every positive integer $n$ , then is it true that $a,b$ are integers ?
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2answers
57 views

Elementary Geometrical Proofs

How do you prove two lines are perpendicular to each other? and then how do you prove that lines ABCD are quadrilateral, or any other specified shape? Is their like a database of certain things you ...
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1answer
70 views

If the average mark of all students in an exam is $30/100$, show at least $20\%$ of the students got at least $10/100$

In an exam the average marks of all the students is $30$ (out of $100$). We have to show that at least $20\%$ of the students have got at least $10$ (out of $100$). I know that it can be shown by ...
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1answer
32 views

The possible value of S

The question is: Find the all possible value of $S(a,b,c,d)= \frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}$ where $a,b,c,d$ are positive. I tried with $a+b+c+d=1$, but it seems I ...
0
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1answer
36 views

Eliminate $p$ from these 2 equations.

$$ X \ = \ 2 \left[ \dfrac {h_1pv_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2pv_2} {(1-p^2v_2^2)^{1/2}} \right] \\ T_2 \ = \ 2 \left[ \dfrac {h_1/v_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2/v_2} ...
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1answer
612 views

Algebra question: Do composite functions remain odd/even

β€œIf $g$ is an even function and $h$ = $𝑓 βˆ˜π‘”$, then $h$ is also an even function.” So tried it with one even function, $g(x)=x^2-5$ and an odd function and $f(x)=5x$ $$f(g(x))=f(g(-x))=25x^2-5$$ ...
2
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2answers
34 views

If $A= \left\{1,2,3,4,5,6\right\}$ and $f(x)$ be a onto fn. from $A$ to $B$ such that $f(i) \neq i \;\; \forall \; i\in \{1,2,3,4,5,6\}$.

If $A,B = \left\{1,2,3,4,5,6\right\}$ and $f(x)$ be a onto function which is defined from $A$ to $B$ such that $f(1) = 2$ and $f(i) \neq i \;\; \forall \; i\in \{1,2,3,4,5,6\}$. $\bf{My\; Try::}$ ...
2
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3answers
84 views

If $f:\mathbb{R}\rightarrow \mathbb{{R}}$ and $f(x)$ satisfying $f(2x+3)+f(2x+5) = 2$ . The period of $f(x)$

If $f:\mathbb{R}\rightarrow \mathbb{{R}}$ and $f(x)$ be a function satisfying $f(2x+3)+f(2x+5) = 2$. Then period of function $f(x)$ is. $\bf{My\; Solution::}$ Let $(2x+3) = t\;,$ Then equation is ...
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4answers
72 views

Why is doesn't $\frac{49^t}{2041}$ = $\frac{(7^t)^2}{2041}$?

Lets assume t = 3 $\frac{49^3}{2041}$ = 57.64 $\frac{(7^3)^2}{2041}$ = 57.64 But according to Khan Academy they're not the same... How is this possible?
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1answer
79 views

Tips on how I would find the transition matrix for the following phenomenon?

how would I go about finding the transition matrix for the following phenomenon (which can be modeled as a Markov process)? Any hints or advice is appreciated! During a study break, a student's ...
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2answers
24 views

Why does $\frac{2}{\frac{1}{8}} \frac{1}{(\sqrt[3]{8})^t}$ equal, $(2 \times 8) \frac{1^t}{2^t}$

$\dfrac{2}{\dfrac{1}{8}} \dfrac{1}{(\sqrt[3]{8})^t}$ = $(2 \times 8) \dfrac{1^t}{2^t}$ Why?
5
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2answers
46 views

The relationship between the intercepts and the remainder in the remainder theorem

The polynomial remainder theorem states that when a polynomial $P(x)$ of degree $> 0$ is divided by $x-r$ ($r$ being some constant) the remainder is equal to $P(r)$, that is: $$\begin{array}l If ...
3
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2answers
82 views

A system of equations

Given three equations $x^2+y^2+xy=a$, $y^2+z^2+yz=b$ and $x^2+z^2+xz=c$, how can I solve for $x,y$ and $z$ in terms of $a,b$ and $c$?
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2answers
39 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 $f(x)$ ...
3
votes
3answers
472 views

If $ab=3$ and $\frac1{a^2}+\frac1{b^2}=4,$ then $(a-b)^2=\;$?

If $ab=3$ and $\frac1{a^2}+\frac1{b^2}=4$, what is the value of $(a-b)^2$? I think $a^2+b^2=36$, please confirm and is it possible to to figure out one of the variables?
3
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2answers
84 views

What is the sum of the power series below?

For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$ What is the sum of it?
0
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2answers
493 views

Parametric equation of ellipse with foci at origin

I want to know what the parametric equation for an ellipse is if the one of the foci is centered at the origin. I know the semi-major and minor axes. I know the parametric equation of an ellipse ...
1
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2answers
158 views

Answer to simple algebraic formula manipulation

I have to show that $2y+(x + 1) = 3 \cdot 2^{x+1}βˆ’ (x + 1) βˆ’ 2$ is equal to $y=3β‹…2^xβˆ’xβˆ’2$ I can get this far: $2y+(x+1)=3β‹…2^{x+1}βˆ’(x+1)βˆ’2$ $2y+(x+1)=3β‹…2^{x+1}βˆ’xβˆ’1βˆ’2$ ...
19
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8answers
3k views

Why do negative exponents work the way they do? [closed]

Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent? $$a^{-b} = \frac{1}{a^b}$$
0
votes
1answer
49 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?
0
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1answer
76 views

What is the outdoor temperature? Working included.

Is my working correct in regards to this question? I'm quite stuck on it and I'm not too sure if I am in the right direction. Any advice is appreciated. Thank you. Question: A thermometer that has ...
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2answers
173 views

Find maximum of a system of equations

You have 300 meters of fencing with which to build two enclosures. One will be a square, and the other will be a rectangle where the length of the base is exactly twice the length of the height. (a) ...
0
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2answers
36 views

Show that $p+q^{2}=1$ where $x^{3}+px+q=0$ and one of the roots is the reciprocal of the other?

let the three roots be $z, 1/z, t$. So $z+1/z+t=0$ and $zt+1+t/z=p$ and $z(1/z)t=-q=t$ $-1/z-z=t$ $p+q^{2}=zt+1+t/z + t^{2}$ How do I simplify the RHS to get 1?
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3answers
66 views

How do I find the sum of the series?

$$\sum_{k=1}^{7}40 \left( \frac{1}{2}\right)^{k-1} = \frac{635}{8}$$ The image of the orginial eqn is on the link above and so is the answer, but I need help in how to solve it. when I did solve it I ...
2
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2answers
13k 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 ...
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1answer
2k views

How to Find the Average Rate of Change (Velocity)?

A problem asked this: The rocket club wants to model rockets straight up into the air. The height of rocket A can being modeled by the function $y=-16x^2+88x$. The function that models the height ...
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5answers
62 views

Where is $(x-1)(x+1/2)\geq0$

The solution is $x\geq 1$ or $x\leq-(1/2)$ However, I do not understand why you flip the sign of $-1/2$ but not $1$.
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3answers
278 views

Moments at which moving points on a circle coincide

Points A $(0,1)$ and B $(1,0)$ start moving along the circumference of a unit circle with center $(0,0)$ in the same, positive (that is, counterclockwise) direction. Every minute, points A and B ...
1
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1answer
39 views

Factoring Trigonometric Equations

And so begins another day in my quest to pass Calculus I. I have a question about factoring trigonometric expressions. I'm sorry I can't be more accurate than that, but I'm not sure of the names of ...
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4answers
1k views

Simple algebra formula for which I can't find the right answer

I have the formula $y + (z + 1) = \frac{1}{2} \cdot (z + 1) \cdot (z + 2)$, and I should work to $y = \frac{1}{2}\cdot z \cdot (z + 1)$. Somebody showed me how it's done: $y + (z + 1) = \frac{1}{2} ...
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3answers
117 views

Simple equation for $x$ but getting no proof.

Show that there is at least one real value of $x$ for which $$x^{1/3} + x^{1/2} = 1$$ I did draw the graphs of $x^{1/3}$ and $1-x^{1/2}$ and showed that they met at a point, but I don't ...
3
votes
5answers
136 views

How do I prove $\frac{ \sqrt{x+h}-\sqrt{x} }{ h}=\frac{1}{\sqrt{x+h}+\sqrt{x}}$?

$$\frac{ \sqrt{x+h}-\sqrt{x} }{ h}=\frac{1}{\sqrt{x+h}+\sqrt{x}}$$ I know I just asked a question and I did figure out how that one worked but I'm not sure how I would go about this one.
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3answers
94 views

How to prove $f(\sqrt{x}) = \sqrt{x}$?

Consider the function $f$ given by $$f(y) = \frac{y+x/y}{2}.$$ How does one prove $f(\sqrt{x}) = \sqrt{x}$? If I plug in the square root of $x$, how do I get back to $\sqrt{x}$? I'm looking over ...
2
votes
1answer
49 views

Steps to calculate, $(\frac{5}{3})^{-3}$

So first I calculate, $(\frac{5}{3})^{3}$ Which gives me, $(\frac{27}{125})$ But where should I go from there?
2
votes
2answers
896 views

Write this surd in its simplest form.

Express $\dfrac{1}{2+ \sqrt3}$ in its simplest form. NB: The textbook has the answer as $2 - \sqrt3$ but I can't see how that was achieved. I tried $\dfrac{1}{2} + \dfrac{1}{\sqrt3}$ and ...
1
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1answer
42 views

Equation of a tangent line to a curve

Equation of the curve is $y=(x+9)/(x+5)$, we are looking for the tangent line to that curve that also goes through $O(0,0)$. Answer given is $x+25y=0$ which I found to be true for $A(-15, 3/5)$ being ...