Lesson: Function Notation

Comment on Function Notation

Hi Brent,

I'm a touch confused with something. I was doing one of the early reinforcement questions and it asked:
If f(x) = 3x − √x and g(x) = x^2, what is f(g(4))?

How exactly do you read out "f(g(4))"? I was confused as to what exactly the question was asking? It seems to me that is asking 'what is f(x) if g(4)?". Is that correct? The problem is that I couldn't see that until I saw your solution in the GMAT Club forums.

Here's the link to the question: https://gmatclub.com/forum/if-f-x-3x-x-and-g-x-x-2-what-is-f-g-209750.html

Thanks for any help,
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-f-x-3x-x-and-g-x-x-2-what-is-f-g-209750.html

The notation f(g(4)) represents the value of the function (f) when we input the value of g(4)

So, in this example, g(4) = 4² = 16

So, f(g(4)) = f(16) [now we must evaluate f(16)]

f(16) = 3(16) − √16
= 48 - 4
= 44

So, to recap, f(g(4)) = f(16) = 44

Does that help?


Awesome! That confirms my assumption. Thanks! :D

Hi Brent,

Could you please help me to solve the following questions taken from a practice test?

For which of the following functions is f(a + b) = f(a) + f(b) for all positive numbers a and b ?

f(x) = x^2
f(x) = x + 1
f(x)= sqrt{x}
f(x)= 2/x
f(x) = –3x

Thank you!
gmat-admin's picture

I'm happy to help.
Here's my full solution: https://gmatclub.com/forum/for-which-of-the-following-functions-is-f-a-b...


Hi Brent,

I'm lost in example:

How does F(x) = F(y)? Was there a mistake in the question and it should have stated F(x) in the last part of the question?

gmat-admin's picture

Question link: https://gmatclub.com/forum/the-functions-f-and-g-are-defined-as-f-x-2x-3...

If f(x) = 2x – 3, then f(7) = 2(7) – 3
And f(19) = 2(19) – 3
And f(5.5) = 2(5.5) – 3
....and f(y) = 2y – 3

The main idea here is that we can input anything into the function named "f"
The function, f, tells us to take the inputted value, double it and then subtract 3.
So, we can also say that....
f(14) = 2(14) – 3
And f(k²) = 2(k²) – 3
And f(4q + 1) = 2(4q + 1) – 3

Does that help?


Hi Brent,

I actually get confused when there are complex functions like these- f(4x) = 8x² − 1

In a simpler equations - f(x) = x^2, we know that the value in the bracket is going to be placed where X is on the RHS.

But in this f(4x) = 8x² − 1, does it mean that the value of x in the LHS will be replaced on the RHS. In short, I am not not able to understand the meaning of such equations.

gmat-admin's picture

I believe you're referring to this super difficult question: https://gmatclub.com/forum/if-f-x-is-a-function-what-is-the-value-of-f-2...

As you suggested, in MOST cases, the function looks something like f(x) = x²
So, f(3) = 3² = 9
And f(-11) = (-11)² = 121
And even f(x + 5) = (x + 5)² = x² + 10x + 25

That said, the function f(4x) = 8x² − 1 isn't much different from the function g(k) = 8k² − 1

To find the value of g(3), we replace k with 3 to get: g(3) = 8(3²) − 1 = 71
However, since f(4x) requires us to multiply the x-value by 4, we must do a little work before we can find the value of f(3)

To find the value of f(3), we need to first find the value of x such that 4x = 3.
Since x = 0.75 is the solution to the equation 4x = 3, we can know that, in order to evaluate f(3), we need x to be 0.75
In other words, f(3) = f[(4)(0.75)] = 8(0.75²) − 1 = 4.5 - 1 = 3.5

Does that help?

Hey Brent,

For this question: https://gmatclub.com/forum/for-all-integers-n-the-function-f-is-defined-by-f-n-a-n-where-a-97735.html

How do we know that we need to square/cube it
gmat-admin's picture

Question link: https://gmatclub.com/forum/for-all-integers-n-the-function-f-is-defined-...

We're told that f(n) = a^n.
So, for example, f(1) = a^1
And f(2) = a^2
And f(3) = a^3
And f(4) = a^4

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