# Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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### Solving an equation with a shift

My professor offered an extra exercise for us to think about. The problem is to solve the shifted equation of the form $$f(x+ia) = x^2 f(x) \, ,$$ where $a$ is a constant. Due to the $x$-dependent ...
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### Computing $\sqrt[3]{1\,}$

I know that the answer is always $1$, but they are looking for some way to get to that answer and I don't know what it is. I am not good at english math terms, but maybe it has to do with differential ...
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### What is the image of $\sin(1/x)\cdot(1/x)$, $0<x<1?$

What is the image of $\sin(1/x)\cdot(1/x)$, $0<x<1?$ I just had a question about what the image of $\sin(1/x)\cdot(1/x)$ is for $0<x<1$. Would it not be all the reals, since $\sin(1/x)$ ...
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### More elegant way to show $X^2+XY+Y^2=Z^3$, if $X=q^3+3pq^2-p^3$, $Y=-3pq(p+q)$, $Z=p^2+pq+q^2$

I believe I have solved the below problem by just expanding the algebraic terms (I will show this), but I am wondering if there is a more elegant way of making the simplification, or if there is ...
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### How solve the problem $f(x+2)=f(x)+4x+4$ for any $x$

Find $f(2012)$, when $f(2)=0$ and $$f(x+2)=f(x)+4x+4$$ for any $x$ I tried to find $f(4), f(6), f(8),....$ \begin{align*} f(4)&=f(2)+4 \cdot 2 +4 \\ f(6)&=f(4)+4 \cdot 4 + 4 = f(2)+4 \cdot (...
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### Is my solution to "MacPOW 1141: Capturing 5 integers" correct?

Crossposted from P.SE as it was closed because I sort of forgot the difference between a math puzzle and a math problem Source: MacPOW 1141 "MacPOW 1141: Capturing 5 integers" states: For ...
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### Function to fit a set of data points [closed]

I am trying to write an equation to describe a line where the values would be as follows. I am so near yet so far. I seem to be unable to paste data from Excel without it coming in as a picture. I ...
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### Problems on lcm and gcd.

Mario has a rest shift every $8$ days; Luigi every $24$ days; Paolo every $16$-th days. Today all three are off. In how many days will all three be back in rest shift for the first time? There are ...
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### Why is subtraction not associative if addition is? [closed]

We know that: $$\forall a,b,c \in \mathbb{R}: (a+b)+c=a+(b+c)$$ Subtraction can be defined as the addition of the additive inverse of a number. So $a-b-c$ can also be written as $a + (-b) + (-c)$ ...
1 vote
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### How do I ensure that I have not lost/gained any solutions when solving a trig/algebraic equation?

Sometimes when I try to solve an equation, I need to multiply by a $\cos(x)$ for example to create a common denominator. Does this create a new solution? Why does/doesn't it? When do I know if when I ...
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### Formula for two overlapping circles, creating three regions with matching area

Suppose we have two circles with radius $r$. The centers of each circle are separated by a distance $x$, such that $0 < x < r$. This creates a Venn diagram where you see three distinct regions, ...
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### If $a,b,c\ge0:a+b+c=3,$ prove $\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$

Problem. Let $a,b,c\ge0:a+b+c=3.$ Prove that $$\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$$ Equality holds at $(1,1,1);(0,0,3).$ I tried to use some classical inequality but it seems ...
1 vote
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### How to algebraically calculate the difference in days for an inverse-proportion or more people=worse problem?

During a drought, 50 people have only $1000 \mathrm{~L}$ of water left. If every person consumes an identical amount of water, the 1000-liter supply would be exhausted in one day. If 40 people were to ...
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### How to determine argument from Euler’s form of complex numbers?

When writing the Euler’s form of a complex number ($z=r e^{i \varphi}$), we say that $r$ is the magnitude of the complex number, while $\varphi$ is its argument, but the same number can be written in ...
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### Finding the max of a function using $f(x)=L(x)-E(x)$ where $L(x)$ is linear and $E(x)$ is exponential

Take this problem, which was on a younger friend's recent AP Precalc test: (note that this test was given by an online provider called "AP Classroom". I highly doubt that this was tested ...
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### Number of solutions for $x ^ y = y ^ x = k$

For a given value of $k \geq 0$, how many solutions $x, y \in \mathbb R$ are there to $x ^ y = y ^ x = k$? My attempt so far: There is the "trivial" solution where $x = y$, and the problem ...
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### What is the linear velocity of the earth in kilometres per hour?

Earth orbits the sun at an average distance of about $150$ million kilometres every $365.2564$ mean solar days, or one sidereal year. What is the linear velocity of the earth in kilometres per hour? I ...
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### Extraneous solutions in algebraic equations

Consider these 2 Equations , where multiplication by $x$ is involved. It is known that multiplication by $x$ should introduce the extraneous solution $x=0$ in both cases. Observe that the second ...
1 vote
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### Find all the polynomials $f$ that satisfy $f(x^2)+f(x)f(x+1)=0$.

find all the polynomial f that satisfies $f(x^2)+f(x)f(x+1)=0$. I'm not sure if I'm doing it in the right way and I'm confused. So I tried to write something like this. let $z_1,z_2,z_3,...,z_n$ be n ...
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### Solution of Two Functions

Given $f(x) = -1 + 5(1.02)^x$ and $g(x) = \ln(3 - x)$, for what value of $x$ does $f(x) = g(x)$? I have been trying to solve this question for quite some time and I always seem to hit a dead end. What ...
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### Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$.

Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ such that $f(x+1)=f(x)+1$ and $f(x^3) = f(x)^3$ for every $x \in \mathbb{Q^+}$. I think this one requires me to prove the linearity of the ...
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### Show $\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$

How can this identity be proved? $$\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$$ I encountered this summation in a probability problem, which I was able to solve using ...
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### How do we prove that $(x^2 + y^2)^2 + x^2 + y^2 < 1$ is a disk?

How do I see that $(x^2 + y^2)^2 + x^2 + y^2 < 1$ is a disk? I plotted it in wolfram alpha and it looks like a disk, but I don't know how to show it algebraically. Even if we write in polar ...
1 vote
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### Volume of A Solid in 3-space consisting of all points (x,y,z) satisfying the Inequality.

Going through serge langs basic math, the section on Area and Applications. I've understood that to find the area of an ellipse of the equation $\frac{x^2}{6} + \frac{y^2}{3}$ = 1 requires one to ...
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### Would appreciate help on finding x according to the equations [closed]

I encountered this question while looking through an old textbook (to which I do not have solutions) in the context of learning about the absolute value function at a grade 11 level. I have ...