# A composition of function problem

1. Jan 27, 2012

### Rron

1. The problem statement, all variables and given/known data
We have f(f(x))=4x-15 , what is f(2)?

2. Relevant equations
Don't know.

3. The attempt at a solution
Don't know how to start actually!

2. Jan 27, 2012

### Curious3141

Let f(x) = ax + b, and see where you go from there.

There are 2 distinct answers here.

3. Jan 27, 2012

### Rron

Curios3141 thanks but couldn't get anything.
Tried a lot.

4. Jan 27, 2012

### HallsofIvy

Then show us what you tried! If f(x)= ax+ b, what is f(f(x))?

5. Jan 27, 2012

### Rron

this is what i tried:
if f(x)=ax+b then f(2)=2a+b
I substituted f(x) in f(f(x)) with ax+b so it becomes f(ax+b)=4x-15
Now ax+b=2 so from this x=2-b\a so if u substitute x in the equation above f(2)=4(2-b\a)-15
then I equalized 4(2-b\a)-15=2a+b, but from this u can't get anything.

6. Jan 27, 2012

### SammyS

Staff Emeritus
Hello Rron. Welcome to PF !

What is f(f(x)), using only the assumption that f(x) = ax+b ?

7. Jan 27, 2012

### Curious3141

As others have already stated, derive an expression for f(f(x)) in terms of a,b and x. Then set this equal to 4x-15 and see what values a and b can take.

8. Jan 28, 2012

### spaghetti3451

I hope no one would mind me writing down the first few lines of the solution to guide our friend here.

Assume that f(x) = ax+b. We need to find f(2), which, from the assumption, equals 2a+b. This means we need to know the values of a and b to find f(2).

To find a and b, substitute f(x) = ax+b in the equation f(f(x)) = 4x-15. You have a[f(x)]+b = 4x-15 i.e. ...

9. Jan 28, 2012

### SammyS

Staff Emeritus
I suppose we should wait for OP to show up in this thread before anyone else posts.

If he doesn't post in a week or two it might be a good idea for one of us to finish this up.

What does the management think about that?

10. Jan 28, 2012

### Rron

Sorry but still nothing. Maybe it is because I learned these a long time ago. It actually has been 3 years since I last solved a function problem like this. So can you please show me the way that you solved this problem in order to save time struggling with the problem and then at the end getting nothing.
Thanks.

11. Jan 28, 2012

### Curious3141

Ah, but if we just present the solution, will you learn anything?

Let's take it step-by-step. It's just algebra.

Then f(f(x)) = a(ax+b) + b = ???

We'll take it from there after you expand the bracket and rearrange terms to get that expression.

12. Jan 28, 2012

### Rron

That's a piece of cake man. a^2x+ab+b

13. Jan 28, 2012

### Curious3141

Great. Now compare that with what the question gave: f(f(x)) = 4x-15

So,

a2x + (ab + b) = 4x - 15

For that to be true in general, the coefficients of each term have to be equal. So you can state:

a2 = 4 ---equation 1

ab + b = -15 ---equation 2

Can you solve that system of simultaneous equations?

14. Jan 28, 2012

### Rron

Curios3141 thank you so much finally solved it.

15. Jan 28, 2012

### Curious3141

It's not the only answer though. Remember a = ±2. b can similarly take different values in each case. You get two perfectly valid forms for f(x). f(2) can take different values depending on which form.

16. Jan 28, 2012

### Rron

Yeah I know that.-1 and 9 but forgot to tell you that I got some choices:
A)-1
B)-2
C)-3
D)-4

17. Jan 28, 2012

### Curious3141

9?

The other possible f(2) should be 11.

f(x) can be 2x - 5 → f(2) = -1

f(x) can be -2x + 15 → f(2) = 11

But here, you go with choice A of course.