Domain and Range of a composite function?? f(x) = x^2+4, g(x) = sqrt/(4x)

markl77

markl77

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I'm given these two original functions :

f(x) = x^2+4
g(x) = sqrt/(4x)

it asks me for the domain and range in interval notation of f(g(x)), which I got [0, +infinity) and (-infinity, +infinity).
This is wrong but I'm not sure when to restrict the values in the range??

and I'm not even sure how interval notation works , is it an x value followed by a y value like this?:
(x,y)
or is it both only x values or y values depending on whether it's talking about domain and range?
 
markl77 said:
I'm given these two original functions :


g(x) = sqrt/(4x)
Click to expand...

You'll have to correct this. This doesn't make sense.

Was it intended to be \(\displaystyle \ \dfrac{\sqrt{x}}{{4x}}, \ \sqrt{4x}, \ \) or something else?
 
greg1313 said:
You'll have to correct this. This doesn't make sense.

Was it intended to be \(\displaystyle \ \dfrac{\sqrt{x}}{{4x}}, \ \sqrt{4x}, \ \) or something else?
Click to expand...

The root 4x , I forgot the notation ...
is it like
sqrt{4x}??
 
markl77 said:
I'm given these two original functions :

f(x) = x^2+4
g(x) = sqrt/(4x)

it asks me for the domain and range in interval notation of f(g(x)), which I got [0, +infinity) and (-infinity, +infinity).
This is wrong but I'm not sure when to restrict the values in the range??

and I'm not even sure how interval notation works , is it an x value followed by a y value like this?:
(x,y)
or is it both only x values or y values depending on whether it's talking about domain and range?
Click to expand...
Interval notation sometime looks like the notation for the co-ordinates of a point on the Cartesian or Argand planes, but it has a different meaning.

Interval notation is specifying a length on the number line. So if you're talking about the domain of
y = f(x), (a, b) refers to values of x whereas if you are talking about the range (c, d) refers to values of y.

The other huge difference is that, in interval notation, you must keep in mind that the form of the brackets is important.

Does this help?
 
g(x) Domain looks good. \(\displaystyle [0,\infty)\). Square roots do that.
g(x) Range is what? Can't get any negative values, can you? This is the first guess at the Domain of f(g(x)). Does f(x) mandate any other Domain Restriction?
You're almost done!
 
markl77 said:
I'm given these two original functions :

f(x) = x^2+4
g(x) = sqrt/(4x)

it asks me for the domain and range in interval notation of f(g(x)), which I got [0, +infinity) and (-infinity, +infinity).
This is wrong but I'm not sure when to restrict the values in the range??

and I'm not even sure how interval notation works , is it an x value followed by a y value like this?:
(x,y)
or is it both only x values or y values depending on whether it's talking about domain and range?
Click to expand...
The domain of g(x) is all non-negative real numbers as is the range of g(x).

Does the domain of f(x) include all of the range of g(x)? Yes, because the domain of f(x) is all real numbers. So the domain of the composite function is simply the most limited of the domains. So I do not see an error in your answer that the domain of f(g(x)) is [0, + infinity).

Now what is the range of f(x)? Its domain is (- infinity, + infinity), but can the RESULT of f(x) ever be negative? Clearly not: the square of a real number is never negative. In fact, can the result of f(x) be 3?

So what do you think the domain of f(g(x)) is?
 
tkhunny said:
g(x) Domain looks good. \(\displaystyle [0,\infty)\). Square roots do that.
g(x) Range is what? Can't get any negative values, can you? This is the first guess at the Domain of f(g(x)). Does f(x) mandate any other Domain Restriction?
You're almost done!
Click to expand...


So if the original range of the first graph f(x) = X2+4 is x is greater than or equal to 4 that range will STAY in the composite function?
I guess that makes sense lol...
 
markl77 said:
So if the original range of the first graph f(x) = X2+4 is x is greater than or equal to 4 that range will STAY in the composite function?
I guess that makes sense lol...
Click to expand...
You have to go back and forth on domain and range of all the functions involved in a composite function to figure out the domain and range of the composite so that it fits all of them at the same time. It's not a cut and paste formula.
 
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