Question

Given that f(x) = 2x + 1 and g(x) = −5x + 2, solve for f(g(x)) when x = 3.

−91
−33
−25
35

Answer 1

Well, you do understand how you would normally solve f(something), yes ?

Let's say you were to solve f(3). What you do when you have to solve that is you look at the form that f(x) has (in this case f(x)=2x+1) and where you see x you replace with 3 and that's how you get f(3)=2*3+1=7.

Answer 2

What you have here follows the exact principle only except that instead of "3" you have g(x).

Answer 3

So it would g(x) = 2(3) + 1 = 7? Im confused :(

Answer 4

And thus when you take what lies between the brackets (in this case we have g(x) instead of 3) and replace it in f(x) you have:

f(g(x)) = 2*g(x) +1

Answer 5

No no no - that was just an example.

Answer 6

Let's try again. If you understand this principle you will no longer have any problem with any such exercise which is why I think is important for you to understand.

Answer 7

Oh okay.
So to solve f(x)
we would do
2(3) + 1, which would equal 7.
and then g(x) would be -5 (3) + 2 and that would equal -13?
so what do we do after that?

Answer 8

No need to calculate f(3), that was only an example to explain the principle behind it.

Here, just ignore the exercise for a minute.

What would you do if you had to evaluate f(5) ?

Answer 9

Let's say f(x) = 2x+1 and I ask you to evaluate f(5).

Answer 10

Im really not sure, I am so horrible when it comes to algebra.

Answer 11

No, it's okay really. If anything it is only a sign of having lazy teachers.

Answer 12

Now, back to our point - when you have f(x)=2x+1 and I ask you to evaluate f(5) what you basically do is

a) Extraction

You take what is between the brackets of what you want to evaluate - so you take what is between the brackets of f(5) which is 5

b) Replacement

You replace what is between the brackets (in this case, the number 5) in the original form of f(x) - in this case, f(x)=2x+1

c) Calculation

You calculate the result, and so f(5) = 2*5+1 = 11

Answer 13

The same thing applies when you want to evaluate f(7) or f(10).

Extraction: 7 and 10
Replacement: 2*7+1 and 2*10+1
Calculation: 15 and 21

Thus f(7)=15 and f(10)=21.

Answer 14

Okay, so I would do that with 3, since it says x= 3?

Answer 15

Now we get to our current exercise which looks a little tricky at first because you no longer have a number between the brackets, you have letters and other funky things.


However, let's try and apply the same procedure and see where that goes.

Answer 16

Extraction: What is between the brackets? g(x).

Okay, we extract g(x). We don't know what g(x) yet, but we follow the same principle we followed earlier. Next up -

Replacement: What is the form of f(x) ? f(x)=2*x+1.

Okay so let's replace x with what we extracted earlier. f(g(x))=2*g(x)+1

Answer 17

Calculation: Well, here we seem to have a problem. We can't really calculate 2 times a letter times a bracket now, can't we ? There's got to be something about it.

Answer 18

We look at the exercise and see that in the description there are two more pieces of information that were given to us - g(x)=-5x+1 is the first piece of information we haven't used so far.

Answer 19

And it does seem to come in handy once we replace in that third step

f(g(x))=2*g(x)+1=2*(-5x+1) + 1

Answer 20

Which is great but still we can't yet give an exact answer to it - and this is where the second piece of unused information comes from.

For x=3 we have that 2*(-5x+1)+1 = 2*(-5*3+1)+1 = -27

Answer 21

I'm sorry if it all sounds confusing but yeah - take what's between the brackets and replace it in the original form of the function. Then take whatever x they give you and replace it in that.

Answer 22

I think I understand, but, -27 isn't an answer choice :(

Answer 23

Oh damn, yeah - sorry about that, g(x) = -5x+2

Answer 24

So that means that f(g(x))=2*g(x)+1=2*(-5x+2) + 1

And for x=3 we have f(g(3)) = 2*(-5*3+2) + 1 = -25

Answer 25

SO much work, holy crap. lol Thank you, I definitely need more practice with this stuff :( Thank you so much for helping, and actually explaining!

Answer 26

No problem!

Once you understand the principle and do 3-4 exercises it will become simple. Take what's between the brackets and replace it in the original form. Then take the form of the function and replace it in that. Then take the value of x and replace it in there. Problem solved.

This can theoretically go on forever.
Like,let's say I gave you 3 functions - f(x)=x, g(x)=x^2 and h(x)=x^3 and ask you to calculate f( g ( h (x) ) ) in x=2. Same principle but 3-4 more lines to write. It looks hard but once you understand the principle it is easy.