What is Precalculus: Definition and 174 Discussions

In mathematics education, precalculus or college algebra is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.

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1. Solve ##\dfrac{3x-6}{5-x}+\dfrac{11-2x}{10-4x}=3\dfrac{1}{2}##

I've multiplied everything out on paper and got ##x=2, \dfrac{15}{4}##, which is correct. However multiplying directly is tedious and from observing this problem I suspect there is a simplification or trick that I missed...
2. Find the time taken by the truck to move ##1## metre.

Unless i am missing something; there is an error with the textbook It ought to be , ## t= \sqrt {\dfrac{1}{0.04}}= \sqrt 25 = 5## seconds.
3. A gardener collected 17 apples...

In fact, it WAS a homework couple of years ago, and I've solved it, kind of (below). I still would like to find a cleaner solution. Here is what I did. Let's say, the apples are labeled, and their weights are ##x_1, x_2, ...##. He takes out the apple #1 and finds that, e.g., ##x_2+x_5+x_9+... =...
4. Question about the property of PDF

This is part of my note: Now, this is practice question: I want to ask why P(X = 2) is not zero, because from the note: P{X = a} = ##\int_{a}^{a} f(x) dx=0## ? If I differentiate F(x), I will get f(x) which is the pdf, then using the pdf to find P(X = 2), I think I will get zero as the...
5. Partial Fraction Decomposition

##\frac {1} {x^2 -c^2}## with ##c \neq {0}## So the first thing I do is split the ##x^2 -c^2## into the difference of squares so ##x +c## and ##x - c## I then do ##\frac {A} {x + c}## ##+## ##\frac {B} {x-c}##, and then let ##x=c## to zero out the expression. And that is where I am getting...
6. B Would like some more knowledge about the product of functions

Hi, PF I only have a clue on the topic I present; the answer involves the ##\subset##, maybe ##\subseteq## concepts; I mean that the only answer I've obtained is that both range and domain of the product of two functions are the inclusion of both. I include a picture regarding the example of...

8. Insights Precalculus, Calculus and Infinitesimals

See my insights article for those interested in an unconventional approach to doing Precalculus at an accelerated pace and beginning Calculus. It is different from the usual way that a precalculus is done text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and...
9. Derivative of a piecewise function

##f(x) = \begin{cases} \frac {x^3-x^2}{x-1} & \text{if } x\neq1 \\ 1 & \text{if } x=1 \end{cases}## Find ##f'(1)##. Using derivative definition: ##f'(1)=\lim_{x \rightarrow 1} {\frac {\frac {x^3-x^2}{x-1}-f(1)} {x-1}}## ##f'(1)=\lim_{x \rightarrow 1} {\frac {\frac {x^3-x^2}{x-1}-1} {x-1}}##...
10. Prove this variation: If a+b ∝ a-b, prove that a^2+b^2 ∝ ab

##a+b \varpropto a-b## means ##a+b = k_1(a-b)## ##(a+b)^2=k_1^2(a-b)^2## ##a^2+b^2+2ab=a^2k_1^2+b^2k_1^2-2abk_1^2## ##2ab(1+k_1^2)=a^2(k_1^2-1)+b^2(k_1^2-1)## ##2ab(k_1^2+1)=(a^2+b^2)(k_1^2-1)## ##a^2+b^2=ab(\dfrac{2k_1^2+2}{k_1^2-1})## Since ##a^2+b^2 \varpropto ab## means...
11. Rounding Errors to nearest 10cm

My answer was 104cm. I was thinking that 105cm would be rounded up to 110cm. The answer in the text book was 105cm.
12. Solve the given problem involving logarithms

Part (a) ##s=ab^t## ##\log s= \log a+ t\log b## Expression on the right hand side increases linearly with ##t##Part(b) ##s=120 ×1.15^t## ##\log s = \log 120+t\log 1.15## ##\log s =2.08+0.06t## From graph, y-intercept = ##2.08## ##m=\dfrac{2.45-2.08}{6-0}=0.06## Part (c)...
13. Solve the given problem that involves binomial theorem

part (a) ##(4+3x)^{1.5} = 2^3+ 9x+ \left[\dfrac {1}{2} ⋅ \dfrac {3}{2} ⋅\dfrac {1}{2}⋅\dfrac {1}{2}⋅9x^2\right]+ ...## ##(4+3x)^{1.5}=8+9x+\dfrac {27}{16} x^2+...##part (b) ##x≠-\dfrac {4}{3}##part (c) ##(8+9x+\dfrac {27}{16} x^2+...)(1+ax)^2 = \dfrac{107}{16} x^2## ... ##8a^2+18a+\dfrac...
14. Solve the given problem that involves Trigonometry

For part (a), We know that ##\cos (-θ)=\cos (θ)## and ##\sin (-θ)=-\sin (θ)## ##\cos (A-B)=\cos A\cos (-B) -\sin A\sin(-B)## ##\cos (A-B)=\cos A\cos (B) +\sin A\sin(B)## ##\cos (A-B)=\cos A\cos B+\sin A\sin B## For part (b) ... ##f(θ)=\cos 60^0- \sin (θ+30^0)\sin (θ-30^0)## ##f(θ)=\cos...
15. Solve the given problem involving functions: f(x)=|ax-b| and y=f(x)+c

A bit confusing here; what i did, Using gradient, we have, ##m=\dfrac{7-1}{0-3}=-2## ##y=-2x+7## Since there is a Vertex, we have the other ##m_2=2## thus, ##y=mx+c## ##1=2×3 +c## ##... y=2x-5## ##a=2, b=-5, c=12## or##a=-2, b=5, c=2##
16. Determine the position vector of ##C##

Highlighted part only... Part (a) was easy ##2\sqrt 5##. For part (b), ...##BC=4i+2j## it follows that, ##OC=OB+BC## ##OC=3i+5j+4i+2j=7i+7j## correct? any other better approach guys! For part (c), I will form the equations as follows; Let ##D(x,y)## then, ##x-4=2(4-3)## and...
17. Find the possible values of angle ##∠ADB##

My take: I got ##BC=10.25## cm, using cosine rule...no issue there. For part (b) ##BK=3cm## using sine rule i.e ##\sin 30^0 =\dfrac{BK}{6}## Thus it follows that ##∠BDK=48.59^0## ...⇒##∠ADB=131.4^0## correct...any other approach? Also: ##∠ADB=48.59^0## when BD is on the other side of the...
18. Prove that these three ratios are equivalent

I have the solution: ## \dfrac{2x-3y}{3z+y} = \dfrac{z-y}{z-x} = \dfrac {x+3z}{2y-3x} ## (1) ## \dfrac{2x-3y}{3z+y} = \dfrac{3(z-y)}{3(z-x)} = \dfrac {x+3z}{2y-3x} ## (2) ##= \dfrac{(2x-3y)-3(z-y)+x+3z}{(3z+y)-3(z-x)+(2y-3x)} ## (3) ##=\dfrac{x}{y}## My question is how is it possible to go...
19. Finding the solution to system of 3 equations with 3 unknowns

For this, I am trying to find solutions, however, I think I am getting a strange result that I am not too sure how to intercept. I first multiply the first equation by 2 to get ##2x_1 - 8x_3 = 4## and then I add it to the second equation below to get ##0 = 1##. I think this means that there...
20. Evaluating the angle theta using inverse sin

I just need to know how to find Θ in sin2Θ=0.51 I know I can use Θ = arcsin(0.51) but what about sin2Θ = 0.51
21. If 1=5, 2=25, 3=125,4=1880, 5=?

I do not know how to do that.
22. Trying to understand the property of absolute value inequality

First lets focus on ##|x|## which is defined as distance between ##x##and ##0##. But if we look into it closely $$13=|-11-2|$$ which is distance between -11 and 2 but $$13=|11-(-2)|$$ which means this is distance between 11 and -2. Which is it? In the same way $$x=|x-0|$$ is distance between 0...
23. Intro Math Wanting to study Functions from scratch for my Calculus preparation

Does Precalculus by James Stewart teaches function from scratch ?
24. Solving Equations: Does Squaring Make False True?

Does this mean for x=1 , ##2(1)-1= -\sqrt{2-1}## is false. x=1 is not a solution. But as we square the above equation , ##(2(1)-1)^2=(-\sqrt{2-1})^2## , false equation becomes true. So now x=1 is solution to the new equation ? (Here is the paragraph attached) from book James Stewart.
25. Problems in manipulating these 4 radical equations

In order to write next step in all four equations above l used the definition of radicals. ##\sqrt a=b## means ##b^2=a##. Squaring both sides also works. I don’t know if it’s right. I mean I read that ##(\sqrt a)^2=a##. But I don’t know if we can apply this on expressions. Main problem is if we...
26. Algebra Algebra 2 textbook recommendations please

I am currently learning some maths from “Precalculus by James Stewart”. I was wondering if that’s ok? Is it ok to just dive straight into it or go back and brush up my algebra 2 ? I was wondering what are some good textbooks on algebra 2 by the way? Thank you. (This is all for the love of physics).
27. Why is 1 not equal to 0 in this proof?

Homework Statement: I am having trouble in understanding the proof. Which step is wrong and why? Relevant Equations: Why ##1 \neq 0## ? Mentor note: Moved from Homework Section ##x=1## ##x^2=x## ##x^2-x=0## ##x(x-1)=0## ##\frac {x(x-1)}{(x-1)}=\frac 0{(x-1)}## ##x=0## ##1=0##
28. Is it Possible for x^2 to Exceed 900?

We can also write a general inequality ##x^2>a## where a is a number. If ##x^2= a##, then ##x= \pm \sqrt a## which means ##x= \sqrt a## or ##-\sqrt a##. But in this case i don’t think it will be ##x > \pm \sqrt a## because if we take ##0## , it’s greater than a negative but in the original...
29. Supplemental topics for precalculus

TL;DR Summary: I will be reviewing/filling in gaps in my precalc knowledge over the course of 6 months and I think I'll have time to go off the beaten path a bit with the material I cover. What topics could supplement this that maybe I would not be able to learn in a typical precalculus...
30. Find all possible solutions of x^3 + 2 = 0

I actually know one way to solve, ##x ^ 3 + 2 = 0## ##x ^ 3 + \left (\sqrt[3] 2\right) ^ 3 = 0## ##\left(x + \sqrt[3] 2\right) \left(x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2 \right)^2\right) =0## ##x + \sqrt[3] 2 = 0, x = -\sqrt[3] 2## ##x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2\right)^2 = 0, x =...
31. Find the values of a real number ##a## for an inequality to hold

Problem statement : Let me copy and paste the problem as it appears in the text. Attempt : From the "Relevant Equations" given above, we can compare to see that ##a-1 = -1## and ##a^2+2=3##. These lead (after some algebra) to the three values of ##\boxed{a=0, \pm 1}##. Issue : The book has a...
32. Intro Math What were the first modern Discrete Mathematics and Precalculus texts?

What was the first textbook for the modern syllabus of precaclulus which had "precalculus" in the title or subtitle? What was the first textbook for the modern syllabus of discrete mathematics which had "discrete," "discrete mathematics" in the title or subtitle? If you have personal...
33. The given function is an even or an odd function?

I think the answer is an even function as the function ##x^2## is an even function and thus, is symmetrical w.r.t. Y axis. The question I have is how to do this problem algebraically. I tried to graph some functions on GeoGebra to verify my answer. a) ##y = ln(x^2)## b) ##y = sin(x^2)##...
34. Courses Precalculus and secondary math study resources

Hey! Now I am also just starting to study secondary math and precalculus, so I am in search of good resources and help. Since I don’t understand a lot of things and I cannot to solve on my own. I have advised you one more resource that I found yesterday. It is called Plainmath, where you can ask...
35. N

Courses Precalculus -- Updating my knowledge

I have been going through the sections in my Ron Larson Precalculus textbook and find that at least for the first 5 or 6 chapters, I can answer a lot of the questions except for a few selective problems. I will post those problems moving forward. No sense in posting questions I already know how...
36. Finding the convergence of a parametric series

It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...
37. MHB Precalculus: What is the value of this sigma notation?

Hi, I'm currently a Grade 11 student and I need help for this question (Precalculus): If $\sum\limits_{i=1}^{50} f(i)=90$ and $\sum\limits_{i=30}^{50} g(i)=60$, what is the value of $\sum\limits_{i=1}^{50} (7 g(i)-f(i)+12)/(2)$? P.S. To those who could answer this, it would be a great help...

This is my attempt so far: ##0.05=\frac{30t}{200000+t}## then I solved for t. And I got 333.88 min. I feel like this is way too simple of a solution and I didn't use all of what's given in the problem. For part 2 of the problem it asks, what happens to the concentration over time. I tried to...
39. Self-Learning a Subtopic within Precalculus

I am attempting to teach myself all the necessary prerequisites of calculus, and attain mastery of them. This has been my goal for quite some time, and recently, I encountered a topic I was unfamiliar with (listed under precalculus). This topic of interest is spherical and cylindrical...
40. MHB Michael Sullivan Precalculus Textbook Arrived Today

I finally got my Michael Sullivan Precalculus 5th Edition textbook today. I have decided to part ways with my David Cohen book. It makes no sense to keep it. It is truly over my head. By this I don't mean that Sullivan's book is a walk in the park. It can be intense depending on the topic...
41. MHB Purchased Precalculus Textbook

Last week, I purchased the Michael Sullivan Precalculus textbook 5th Edition, a book I had in my collection in 2006. I will use this book to continue my precalculus journey as I make my way to calculus 1. Anxiously waiting for the textbook to arrive in the mail. I will stop by the post office...
42. Foundations Precalculus: Sixth Edition by Warren W. Esty, is it Rigorous?

Hi PF, I happened up a textbook called Precalculus (Sixth Edition) by Montana State University Professor Emeritus, Dr. Warren W. Esty and I understand that it is supposed to be a great textbook book for Precalculus self-study. I was just wondering if anyone had heard of, or used it in the...
43. What do I need to study before precalculus?

Summary: I need to know what to study in order to prepare for precalculus I'll be starting classes in spring and I'm working towards a bachelors in computer science. My first math class will be precalculus. Math has always been an intimidating subject for me(my WEAKEST subject), but I'm...
44. Other Serge Lang's Basic Mathematics vs David Cohen's Precalculus

Hello, as the title states, which book is better for learning Precalculus? I heard David Cohen's is much more application/computational based and is better for learning Precalculus rather than Lang's. However, Lang is also good as well because he doesn't focus on plug and chug, but rather...
45. Calculating the half maximum point of a function

This is the form of the function above: I started by equating (1) to 1/2: $$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$ which can be rearranged to: $$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$ using...
46. Find the derivative of a balloon's circumference

A balloon's volume is increasing at a rate of dV/dt. Express the rate of change of the circumference with respect to time (dc/dt) in terms of the volume and radius. Homework Equations Vsphere = (4/3)(π)(r^3) C = (2)(π)(r) The Attempt at a Solution [/B] My strategy was to come up with two...
47. Hi, I have a quick question about graph transformations.

One of my homework questions said "Explain how to obtain f(x)=-(3+x)^2+1 from the graph of y=x^2." I know somehow you need to move the graph right 3, reflect about the x-axis, and move up one, but I don't know how to factor and manipulate the equation to show this.
48. Find the center of a circle given a tangent line & point

Homework Statement "Find the center and radius of the circle that passes through A(1,1) and is tangent to the line y=2x-3 at the point B(3,3). (Picture of the graph: https://imgur.com/a/0wAnqcU) Homework Equations Here's a link: https://imgur.com/a/y71Z9GY The Attempt at a Solution Soo, I've...
49. Hi, I have a quick question about inverse functions.

One of our homework problem asks: If f is a one-to-one function such that f(-3)=5 , find x given that f^-1 (5)=3x-1. Here's how I attempted to solve the problem: -3=3x-1 3x=-2 x=-2/3 Is this the correct way to solve the problem?
50. How Do You Solve Matrix Equations with a Calculator?

Homework Statement Homework EquationsThe Attempt at a Solution I know I would have to do something with my calculator and I tried to solve like solving an equation for C, but not sure. I put all the matrices in my calculator. I then subtracted the first matrix to the other side then...