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
2answers
73 views

Evaluating $\lim_{x\to1}\frac{x^2}{(x-1)^2}$

$\lim_{x\to1}\frac{x^2}{(x-1)^2}$ How do I calculate this limit? I can't use L'Hospital's rule here and dividing by $x^2$ doesn't work either. Thanks
0
votes
2answers
28 views

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$ I tried to factor and I got $2\cos^4(x)+(-2\sin^2(x)-3)(\cos^4(x)+\sin^4(x))$ but that doesn't lead me to my goal. ...
0
votes
1answer
12 views

Polynomial Functions and Their Graphs - Seeking Help for a Deeper Understanding

Match the the polynomial function with one of the graphs I - IV. Give reasons for your choice. 1) P(x) = x(x^2-4) I chose number III because it reminded me of a power function raised to the 3rd ...
2
votes
3answers
213 views

Reason what 2^k in a sigma will do

I am trying to solve the following calculation without a calculator: $$\sum_{k=0}^82^k{8\choose k}$$ The first part: $$\sum_{k=0}^8{8\choose k}$$ is equal to $2^8$. I already know that the answer ...
2
votes
1answer
58 views

Sketch the Graph. $S(x) = \frac{1}{2} (x-1)^3 +4$

The question is: Sketch the graph of the the function by transforming the graph of an appropriate function of the form $y = x^n$. Indicate all x and y intercepts on the graph. I am really trying to ...
1
vote
3answers
35 views

Sketch the Graph of $-x^3+27$

Not sure if this is the right way to solve. The question ask to sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^n$. Indicate all $x$- and ...
0
votes
2answers
39 views

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$ I really am stumped on this problem. I squared the first equation and found that $-y = \sqrt{(x+y)(y+z)}$. So ...
2
votes
5answers
59 views

How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$?

How to find the coefficient of $x^8$ in $(x-1) (x-2) . . .(x-10)$. Is there any general formula to solve this kind of problems?
0
votes
1answer
28 views

extrema of a polynomial

Say I'm given a general sextic equation $$ax^6+bx^5+cx^4+dx^3+ex^2+fx+g =0$$ I want to know how I can tell where the critical points (or least say how many there will be) will be since the ...
0
votes
2answers
36 views

how do i write an equation in standard form by completing the square for $x^2 -9y^2-4x-18y=14$

I'm really having trouble with completeing squares i can solve for circles and ellipses but i can't seem to understand hyperbolas or parabolas, help would be deeply appreciated.
0
votes
3answers
57 views

Sketch the graph of the polynomial function $P(x)= x(x-3)(x+2)$

Sketch the graph of the polynomial function. Make sure your graph shows all the intercepts and exhibits the proper end behavior. $$P(x) = x(x-3)(x+2) $$ Should I ignore $x=0$ as a value?
0
votes
1answer
12 views

Using Zeros to Graph Polynomials - Loose Definition Needed for Comprehension

I am having a hard time understanding this definition. The definition states. "If P is a polynomial function, then "c" is called a zero of P if P(c) = 0. In other words, the zeros of P are the ...
2
votes
8answers
130 views

How to solve $x$ in $(x+1)^4+(x-1)^4=16$?

I'm trying my hand on these types of expressions. How to solve $x$ in $(x+1)^4+(x-1)^4=16$? Please write any idea you have, and try to keep it simple. Thanks.
1
vote
4answers
49 views

how to find the root in $x^3-x^2+6x+24$?

I'm trying my hand on these types of expressions. how to find the root in $x^3-x^2+6x+24$ ? please write any idea you have and easy ways please ! thanks.
0
votes
1answer
35 views

Reversing $\sum_0^kx\cdot2^i$

I'm trying to write an algorithm, and the easiest way I found for explaining the mathematical problem I'm facing is the following: Assume you have $x$ US dollars and you're gambling on a roulette. ...
1
vote
3answers
29 views

Two Questions Regarding Polynomial Functions.

I am studying Polynomial functions and their Graphs. I am currently looking at the definition for a polynomial function and I am trying to arrive at a deeper understanding; thus, please excuse ...
0
votes
3answers
26 views

Quadratic Function is Given - x^2-4x+4. What are the x intercepts

I am having a problem finding the x intercepts. I have found the Standard form of the Quadratic (Vertex form) -(x+2)^2 + 8 [by the way, does is matter if the 8 is in front of the minus sign or in ...
7
votes
1answer
135 views
+50

How prove this equation roots $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

Question: Let $a,b,c>0$ are give numbers and $x>0$, such that $$ \sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}} $$ show that $$ ...
1
vote
3answers
80 views

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

This is supposed to be an application of AM-GM inequality. if $abc=1$, then the following holds true: $a^2+b^2+c^2\ge a+b+c$ First of all, $a^2+b^2+c^2\ge 3$ by a direct application of ...
2
votes
3answers
37 views

How do I find the $x$ intercepts for $-x^2-3x+3$

How do I find the $x$ intercepts? $-x^2-3x+3$? I converted the function into vertex form but I am stuck at $3/4= -(x^2+1.5)$. Can someone give me and idea what I can do?
0
votes
3answers
35 views

Work out linear factors

$x^3 + 2x^2- 9x -18 = (x^2 - a^2)(x+b)$ were $a$ and $b$ are integers. Work out the three linear factors of $x^3 + 2x^2 - 9x - 18$. What would I do to answer this? I don't have a clue where to start ...
2
votes
3answers
116 views

How to solve this equation? $y^4-2y^2-\sqrt{y+1}=0$

How to solve this equation? $y>0$ $$y^4-2y^2-\sqrt{y+1}=0$$ I found one solution that $y=\frac{1+ \sqrt5}{2} $ Can someone help me?
0
votes
1answer
32 views

Find $t$ in $N = b \times g^t$.

The problem is the following: Find the value of $t$ in $N = b × g^t$. So for example "$512.000 = 2000 × 2^t$" I'm not really a mathematician so their may be a simple way or it could be hard.
0
votes
1answer
19 views

$f(x) = 2x^2 + 4x+3$, Express in Vertex Form: Give the Vertex and $y-$ and $x-$ intercepts

I can't find the $x$ intercepts! My vertex form is $2(x+1)^2 +1$ My $y-$intercept is $3$ and my vertex is $(-1,1)$ However, I am stuck trying to find the $x$ intercepts. I have $-\frac{1}{2} = ...
1
vote
2answers
35 views

Parabola problem involving depth.

The equation for the parabola, I found, is $$y^2=3.2x$$ What is the depth of the reflector? I'm getting $\dfrac {25}{3.2}$.
1
vote
1answer
45 views

I Completed the Square for x^2-6x, but I don't have the correct x intercepts. How do I find the roots?

x^2 -6x when converted into standard form is (x-3)^2 -9 It has a vertex of (3,-9) It has a y intercept of y = 0 However, I thought the x intercept would be +3. Yet, the book says the x intercepts ...
-1
votes
0answers
39 views

Parabola word problem. Depth? [duplicate]

I did the first part of this problem correctly, coming up with the equation of y^2=3.2x However, for the second part, where it wants to know the depth of the reflector, I’m having trouble. I ...
0
votes
3answers
38 views

Seeking a deeper Understanding of Inverse functions f(x) = 4 - x^2, x> or = 0

I am seeking a deeper understanding of inverse functions $$f(x) = 4 - x^2, x\geq0$$ How does the Domain affect how one evaluates the function?
0
votes
0answers
35 views

Question about an exponential funtion

Now we have a function: $f(x)=e^x, x\in \mathbb R$ Question: 1) Assume that $x>0$, discuss the number of the intersects between $f(x)$ and $y=mx^2,m>0$ under different situation. 2)Assume ...
0
votes
2answers
35 views

value of $x^3 - 6x^2 + 6x$ when $x$ is an expression involving radicals

If $x = 2 + 2^{\frac13} + 2^{\frac23}$, then what is the value of $x^3 - 6x^2 + 6x$? How would I solve it? Surely plugging in is one way, but there's got to be some other way.
3
votes
1answer
41 views

How to solve $(1-\tan^2x)\sec^2x+2^{\tan^2x}=0$

I am stuck the following problem : Solve the following problem: $(1-\tan^2x)\sec^2x+2^{\tan^2x}=0$ Taking $p=\tan^2x,$ we get $(1-p^2)+2^p=0$. Now,I do not know how to go from here. ...
0
votes
2answers
54 views

Find the inverse function of $ f(x) = 4 - x^2$,$ x\geq 0$

Find the inverse function of f $$ f(x) = 4 - x^2, x\geq 0 $$ I know that Power Functions raised to the second power are not one to one; however, since the domain for this function is $[0,\infty)$, it ...
0
votes
3answers
30 views

Find the inverse function of $f$. How do I isolate the $y$ if $f(x)= x/(x+4)$?

Find the inverse function of $f$. How do I isolate the $y$ if $ f(x) = x/(x+4)$? I am stuck at $(y+4)x = y$. I can't divide both sides by $y$. I am stumped.
0
votes
0answers
12 views

Finding the appropriate angular velocity given the velocity of the intersection of the bars and other parameters

Suppose we a "scissor", as you close the bars together with a constant angular velocity $\omega$, I came up with this equation:$$\frac{-\omega}2\frac{AO}{tan\frac{{\omega}t}2}=v$$where "v"is the ...
1
vote
3answers
39 views

Finding the fourth vertex of a square

I was given this problem: Without drawing a graph and given the following 3 vertices, find the coordinates of the last vertex of the square: $(3,2)$$(0,5)$$(-3,2)$ So first, I found the ...
0
votes
2answers
29 views

Extracting equations from a word problem

The difference of two integers is $18$. One integer is one-third of the other integer. What are the integers? I know that the first equation is $x-y=18$, after that..was confused.
0
votes
2answers
34 views

Completeness of the real numbers

I'm fairly new to formal proof, so when I started learning about real analysis it has been a huge source of confusion to me. Not too long ago I was introduced to the least-upper-bound property, or, ...
1
vote
3answers
40 views

I'm stuck. Can somebody help me solve this problem.

The sum of our present ages is $41$. If I live $17$ more years, after doubling my present age, I will be $9$ years less than your present age. What are our present age?
0
votes
2answers
29 views

Finding the fifth term of this arithmetic progression problem

If the sum of the first 10 term is equal to $-240$ and the 7th term is $48$. How to find the 5th term?
0
votes
1answer
30 views

Transformations of Functions - What are the Order of Operations?

I would like to confirm the order of operations when it comes to transformations. For example: 1/2 square root (x+4) -3 Is the order of operations: Evaluate expression within the parentheses 1st ...
1
vote
3answers
26 views

Basic Algebraic Manipulation

How would I solve for $X$ in this instance? I can't figure out how to get the $X$ variables by themselves and the known values on the other side by themselves. $2(4-X)(4-X)+X = 3$
3
votes
3answers
84 views

What does $f(x)$ denote?

All I have been reading from my textbook, and I am wondering if my textbook is just badly written since I have learned in the past that $f(x)$ means $y$ (output) and the variable $x$ is the input. ...
0
votes
1answer
23 views

Proving that $-\frac{\cos x +\cos^2 x + \sin^2 x}{(1 + \cos x)^2} = -\frac{1}{1 + \cos x}$

Could anyone give me some details on how this happens? $$-\frac{\cos x +\cos^2 x + \sin^2 x}{(1 + \cos x)^2} = -\frac{1}{1 + \cos x}$$
1
vote
1answer
44 views

Solve this simple polynomial

$$\text{Problem: }{x^3-x^2-x-2 \over x^2 + x - 6}$$ My textbook was able to come up with $(x-2)(x^2+x+1)$ $$\text{Textbook: }{(x-2)(x^2+x+1) \over (x-2)(x+3)}$$ I've tried grouping and using the ...
0
votes
0answers
14 views

How many hops plants can I plant in a circle around a central point?

I need to plant hops rhizomes in a circle at a distance of 10.5 feet from a central pole, at distances of 3 feet from each other. How many hops rhizomes can I plant around my pole?
2
votes
5answers
269 views

answer to the $(x+1)(x+2)+(x+3)(x+4)+\cdots+(x+99)(x+100)$? [closed]

I'm trying my hand on these types of expressions.how to answer to the $(x+1)(x+2)+(x+3)(x+4)+\cdots+(x+99)(x+100)$ ????? please write any ...
0
votes
7answers
71 views

how to solve x in $2x^4-3x^2+1=0$?

how to solve x in $2x^4-3x^2+1=0$ ? or any idea how to turn it into a Quadratic equation?
0
votes
3answers
44 views

Determine whether the Question is One-to-One; f(x) = 3x+4?

I am having a problem understanding how to determine if a function is one to one. The problem is: Show that the function f(x) = 3x+4 is one-to-one. Also, I'm being thrown off by the notation ...
1
vote
5answers
46 views

determining the maximum of a rational function

I was going over some practice problems and got stuck with this one: I am supposed to find the maximum of the function: $$\dfrac{x}{x^2+1}$$ on the interval $(0,4)$. All I can think of is to ...
0
votes
1answer
47 views

Is there a term for this subtraction formula?

Is there a term for this concept? Any link? n is a decimal from 0 to 1, including FORMULA n = 0.5 x = 1 - n EXAMPLES ...