Main content

# Subtracting functions

Given that f(x)=2x√5-4 and g(x)=x^2+2x√5-1, Sal finds (g-f)(x). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- What does "the principle square root" mean? Sal wrote it normally as sqrt of 5, but I'm not sure what the "principle" means here, since I have always heard square roots read aloud as "square root of 5" or whatever number you're taking the square root of. Please help! Thank you.(19 votes)
- The principle root of a number is just the positive square root. Each number has 2 square roots. For instance, the square root of 36 could be either 6 or negative 6. The principle root is 6.(1 vote)

- at1:07Sal talks about the principle square root of 5. What is a principle root? Is there a vid on it.(2 votes)
- The principal square root is the positive square root. There are 2 different numbers that can be squared to make 5, +sqrt(5) and -sqrt(5). If you squared either, they would make positive 5.(14 votes)

- At0:04, Sal refers to the square root 5 as the principle square root of 5. Why is this?(2 votes)
- A square root generally has two answers. A positive answer and a negative answer.

A square root is asking, "what number, times itself, gives me this number?" So the square root of 25 is asking, "What number, times itself, gives me 25?". Well... 5*5 gives me 25. However, -5 * -5 also gives me 25. Both are valid answers for the square root of 25.

In general, most of the time, you want a positive number as your answer to a square root. The positive possible answer is also known as the "principal square root". So the principal square root of 25 is 5.

Sal just wanted to make it clear that he is using the positive possible answer, and not the negative possible answer to the square root of 5(4 votes)

- i didnt understand any of it...'(3 votes)
- The principle root of a number is just the positive square root. Each number has 2 square roots.(3 votes)
- I don't get it. The video won't play for me. :((2 votes)
- Check to make sure your computer could play a video.

I never have this problem on KA Website.(2 votes)

- I'm confuse with the "removing the pharanthesis" part:

On the video, if we're going to substitute g(x)-f(x) with its definition, it will become:

(x^2+2x√5-1) - (2x√5-4). To get rid of the parenthesis, we have to multiply an imaginary -1 to the exprission, 2x√5-4. So, -1(2x√5) = -2x√5 and -1(-4) = 4, right? But why does the video say that -1(-4) = 5?(2 votes)- Hi! Good question. They do multiply by a -1 and that changes the numbers making a (-) a (+). So, that's why they removed the parenthesis. I don't know if this helps but I hope it does.(1 vote)

- so I'm 12, and I'm new to this, so do not judge me just help me, but for some reason, this is what I did:

(x^2+2*√5-1)-(2x√5-4)

(x^2+2*0-1)-(2x*0-4)

(x^2-1)-(-4)

(x^2-5)

but the answer was x^2-3 what did I do wrong.(2 votes)- First, you changed the expression. It should be:

(x^2+2x√5-1)-(2x√5-4)

Next - Where did you get the 0's from in your 2nd line? You can't change √5 into 0. They are not equal.

The correct way to work the problem is to distribute the minus sign that is in front of the 2nd set of parentheses. This minus is the same as -1 just like "-x" is the same as "-1x". Distributing the minus or -1 results in changing the signs of the terms inside the parentheses:

(x^2+2x√5-1)-1(2x√5-4) = x^2+2x√5-1-2x√5+4

Then, you combine like terms:

x^2+2x√5-1-2x√5+4 = x^2+(2x√5-2x√5)+(-1+4)

= x^2+3

You never applied the minus sign to the -4. Instead, you treated: "-(-4)" as just "-4".

Hope this helps.(2 votes)

- What does Sal mean by "f of x"?(2 votes)
- "f of x" refers to function notation. In this video, you are working with functions. You should have gone thru the lessons on what is a function and what is function notation much earlier than this video. See this link for a quick overview: https://www.mathsisfun.com/sets/function.html(2 votes)

- This video was really helpful, I just had a question. Do they always give you the values for the functions (f of x and g of x) or do you sometimes have to find them yourself? Thanks.(2 votes)

## Video transcript

- [Instructor] We're told
that f of x is equal to two x times the square root of five minus four. And we're also told that g
of x is equal to x squared plus two x times the square
root of five minus one. And they want us to find g minus f of x. So pause this video, and see if you can work
through that on your own. So the key here is to just
realize what this notation means. G minus f of x is the same thing as g of x minus f of x. And so, again, if this was helpful to you, once again I encourage
you to pause the video. All right, now let's
work through this again. So this is going, or,
I guess the first time, but now that we know that this is equal to g of x minus f of x. So what is g of x? Well, that's the same thing
as x squared plus two x times the square root of five minus one. And what is f of x? Well, it's going to be two x times the square root of five minus four. And we are subtracting f of x from g of x. So let's subtract, this
is f of x, from g of x. And so now it's just going to be a little bit of
algebraic simplification. So this is going to be equal to, this is equal to x squared plus two x times
the square root of five minus one. And now we just have to
distribute this negative sign. So negative one times two x times the square root of five is, we're gonna have minus two x
times the square root of five. And then the negative of
negative four is positive four. Now let's see if we
can simplify this some. So this is going to be equal to, we only have one x squared term, so that's that one there. So we have x squared. Now let's see, we have two x
times the square root of five. And then we have another, oh, and then we subtract two x
times the square root of five. So these two cancel out with each other. So those cancel out. And then we have minus one plus four. So if we have negative one
and then we add four to it, we're going to have positive three. So if we just fact, if we take this and
this into consideration, four minus one is going
to be equal to three, and we're done. That's what g minus f of x is equal to, x squared plus three.