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## SAT

### Course: SAT > Unit 10

Lesson 2: Passport to advanced mathematics- Solving quadratic equations — Basic example
- Solving quadratic equations — Harder example
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems — Harder example
- Manipulating quadratic and exponential expressions — Basic example
- Manipulating quadratic and exponential expressions — Harder example
- Radicals and rational exponents — Basic example
- Radicals and rational exponents — Harder example
- Radical and rational equations — Basic example
- Radical and rational equations — Harder example
- Operations with rational expressions — Basic example
- Operations with rational expressions — Harder example
- Operations with polynomials — Basic example
- Operations with polynomials — Harder example
- Polynomial factors and graphs — Basic example
- Polynomial factors and graphs — Harder example
- Nonlinear equation graphs — Basic example
- Nonlinear equation graphs — Harder example
- Linear and quadratic systems — Basic example
- Linear and quadratic systems — Harder example
- Structure in expressions — Basic example
- Structure in expressions — Harder example
- Isolating quantities — Basic example
- Isolating quantities — Harder example
- Function notation — Basic example
- Function notation — Harder example

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# Function notation — Harder example

Watch Sal work through a harder Function notation problem.

## Want to join the conversation?

- I learnt from this video but the practice questions I solved were completely different from the question solved in the video.

How to solve the questions of function notation related to graph??(58 votes)- just give the some inputs to given function and take output values then mention them on graph(1 vote)

- Why wasn't the last answer an option? It looked equivalent to the equation before it was simplified.(6 votes)
- The last answer İSN'T equivalent to the equation before it was simplified.

As you can see at1:20there is a "cube root sign surrounding (x^3+1)" inside another cube root sign outside. İf this inside cube root sign didn't exist around (x^3+1), then you would be right.

At1:24we're raising this "inside cube root of (x^3+1)" to the third power, that's why they cancel each other, and we're left with (x^3+1), add 1 to that and you get the correct answer.(21 votes)

- where can i get more practice on function notation please?(18 votes)
- Is the SAT practice hard if so scale it one to 20 ?!(5 votes)
- It depends per person based on practice level, familiarity with concepts etc... for me it was a 10-11 (without all the decimals lol)(2 votes)

- I need help solving u^2 - 125=0(1 vote)
- Your first step is to look for factors that you can use to solve the equation quickly. Here, I'm not seeing anything, so we'd go to one of the methods for solving quadratic equations that can't be factored: the quadratic formula and completing the square. Completing the square is generally faster when you have just the a and b terms, or if the numbers are easy to work with. I don't really think it'll make much of a difference here, so we'll use the quadratic formula.

x = -b +/- sqrt(b^2 - 4ac) / 2a

x = -0 +/- sqrt(0 - (4)(1)(-125)) / 2(1)

x = +/- sqrt(500) / 2

x = +/- 10/2 * sqrt(5) = +5sqrt(5) and -5sqrt(5)(5 votes)

- can't option B also be the answer? it states the exact same thing only in an expanded form.(1 vote)
- On the SAT, you should be careful about parentheses and order of operations in general. Here, notice that in option B) the last +1 is not under the radical, which means that it is different than option C) because you can't combine like terms under an exponent like you maybe could with multiplication.(4 votes)

- “This is gonna be fun!”
**cue nervous laughter**(2 votes) - what do you have to put in mind as you do such questions on the SAT(2 votes)
- I do not know that(1 vote)

- why not the second choice ? it's equivalent to the third.(2 votes)
- cbrt(x³ + 1) + 1 ≠ cbrt(x³ + 2)

https://www.desmos.com/calculator/cbxo25elpy(3 votes)

- Can this answer further be simplified? Can we take x^3 under the radical separately and root 2 separately and then the answer would be x +or- root 2 ?(1 vote)
- Of course not. When there's addition of subtraction inside square roots (or any roots, really), you can't take the root of one value then take the root of another value then add them together.

Meaning:

sqrt(a + b) =/= sqrt(a) + sqrt(b)

sqrt(a - b) =/= sqrt(a) - sqrt(b)

However, when you have multiplication or division, then you can break them into other parts.

sqrt(a * b) = sqrt(a) * sqrt(b)

sqrt(a / b) = sqrt(a) / sqrt(b)

(sqrt is short for square root of, and =/= means "not equal to")(2 votes)

## Video transcript

- [Teacher] Let y of x be equal to the cube root
of x to the third plus one. Which of the following is
equivalent to y of y of x? All right, this is gonna be fun. So we know that y of x is equal to the cube root
of x to the third plus one. Which of the following is
equivalent to y of y of x? Well, to evaluate y of y of x, y of, y of y of x, everywhere where we see an x here, we will just replace it with a y of x. And it's kind of, it might seem a little bit daunting 'cause it's referring to itself but it should all make sense in the end. So, we're gonna have something
to the third power plus one, and in this case, that
something is y of x. y of x. So, what's y of x equal to? Well, it's equal to all of these business. So, let me write this down. So this is going to be equal to the cube root of, let me give myself some space. So, y of x is the cube root of x to the third plus one. Now we're going to raise
that to the third power. We'll raise that to the
third power plus one. And so we just have to evaluate this. So, what is this going to be? Well, this is going to be the cube root of, well, what we have right over here. If I take the cube root of something but then I raise it to the third power, this is like raising something to the 1/3 and then raising that to the third. Well, that's just gonna
give me this thing. This, all of this
business right over here, is just going to evaluate
to x to the third plus one. Once again, if I just take the cube root of something, of whatever this is, and then raise that to the third power, that's going to be this
original something. Maybe I should put a, if I take the cube root of a and if I raise it to the third power, that's just going to be a here. That's all I did right over here. The cube root of x to the third
plus one to the third power is just gonna be x to the third plus one. And then now, we have the plus one, so this is going to be
equal to the cube root of x to the third plus two. x to the third plus two, which is that choice right over there.