Combining functions and Domain

1. Apr 20, 2013

ThomasMagnus

1. The problem statement, all variables and given/known data

F(x)=√x

G(x)= 2/(x-4)

Find g(f(x)) and its domain

2. Relevant equations

3. The attempt at a solution

I am having trouble finding the domain for this composition. I know that g(f(x))= 2/(√x)-4

I believe that for the domain you combine the domain of the input (f(x)) and the output (g(f(x)). This would then be x > 0 (for f(x)) and x≠16 in g(f(x)). The answer key that I have says that the answer is x>0, x≠4

Where am I going wrong? Why does it disregard 16?

Thanks!

Last edited by a moderator: Apr 20, 2013
2. Apr 20, 2013

Dick

You aren't going wrong. The answer key is wrong. Though x could be zero, right?

3. Apr 20, 2013

ThomasMagnus

Woops, I should have put x≥0.

So for these types of problems, the domain of the combined function will be the domain of the input and the composed function? Why is the other ignored? i.e why can x be 4 if it will make g divide by 0?

4. Apr 20, 2013

Dick

You didn't ignore the 4. That's why you said 16 isn't in the domain, right? Yes, it's the domain of g(f(x)) you want.

5. Apr 20, 2013

ThomasMagnus

But why can x be 4 if it makes g(x) undefined?. Don't I have to also say x≠4?

6. Apr 20, 2013

SammyS

Staff Emeritus
You're working with g(f(x)).

Therefore, if g(4) is undefined, then g(f(x)) is undefined when f(x) = 4.

For what value of x is f(x) = 4 ?

7. Apr 20, 2013

Dick

g(f(4))=g(2)=(-1). It's not undefined at all when x=4. It's undefined at x=16, when f(16)=4 and g(4) is undefined. Or it's undefined when x<0 when f(x) is undefined. As you said to begin with.

8. Apr 20, 2013

ThomasMagnus

Aha 16. So, with compositions of functions you're paying attention to the input (in this case f(x)) and the final compositions (g(f(x)). So the domain of the whole thing is pretty much the input and composed function domains combined?

Thanks for all the help! :)