Problem Help!

GRE Quant Problem Solving
1

Why does the problem say “f(a)=f(b)=0”? It completely confused me. On Greg’s explanation video there’s is no mention of them.

f(x) is a quadratic function such that f(a)=f(b)=0, which of the following are the roots of f(2x)?

Choose two answers.

  • 0
  • a/2
  • b/2
  • a
  • b
2

f(x) is a function. Think of a function like a black box which gives some output based on some input.

Let’s take one such function, where f(1)= -1, f(2)=-2, f(3)=-3, f(-1)= 1, f(-2)=2, f(-3)=3… and so on. You can see whatever you give as an input, the function is changing the sign. so for any x you give as an input, f(x) = -x. This will be the definition of the function.

Similarly, let’s assume some random function whose equation (what happens inside the black box) we don’t know. It says f(a) = 0. Meaning if you give a as the input, the black box spits out 0 as the output. Similarly, f(b)=0 also means that for an input b, output of the function will be 0.

“Roots of something” is a fancy way of saying the same thing. It means find the value of the variable, so that the output is 0. Roots of a quadratic equation means the values of x which when substituted in the equation will add up to 0. Roots of a function means the values of x such that the output of the function is 0.

3

Thanks for the reply. I still don’t get why the “f(a)=f(b)=0” was needed to answer the question.

4

Okay, so if that wasn’t given, which two answers would you choose?

5

I’m confused tbh. This is what Greg says on the solution video:

Solving with an example:

  • Create an example were f(x) = x^2-1
  • Passing an x results in x = 1 and -1
  • Passing a 2x results in 1/2 and -1/2
  • Conclusion: all the function does is divide by 2 therefore answers are a/2 and b/2

Solving conceptually

  • Since it’s asking you for the roots (where the parabola crosses the x-axis) and it’s asking you for 2x (which makes the parabola skinnier), the only possible answers are a/2 and b/2.

This is why I’m so confused as to way “f(a)=f(b)=0” is important.

6

Both of the methods you have used to solve this problem are absolutely correct.

But for your doubt about the importance of a and b, try to answer this now.

f(x) is a quadratic function, which of the following are the roots of f(2x)?
Choose two answers.

  • 0
  • Θ/2
  • ρ/2
  • Ψ
  • β

I’ve just put random symbols to say that a and b don’t have any significance since it’s not at all mentioned in the question.

7

Thank you! That answers my question.