Combining functions and Domain

[ThomasMagnus]

Homework Statement

F(x)=√x

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

Find g(f(x)) and its domain

Homework Equations

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!

 

[Dick]

Homework Statement

F(x)=√x

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

Find g(f(x)) and it's domain

Homework Equations

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!

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

 

[ThomasMagnus]

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

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?

 

[Dick]

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?

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.

 

[ThomasMagnus]

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.

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

 

[SammyS]

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

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 ?

 

[Dick]

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

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.

 

[ThomasMagnus]

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 ?

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! :)