# Thread: Range of composite functions

1. ## Range of composite functions

Given two functions:

f(x)=sqrt(4-x)
g(x)=x^2

Find the range of gof:
gof(x)=4-x

Shouldn't the range be all real numbers? Because it is a linear function. But the answer key says [0,infinity[ ??

Next question:

Given two functions:
f(x)=1/x (x cannot be 0)
g(x)1/x^2 (x cannot be 0)

Find the range of fog

fog(x)=x^2

Shouldn't the range by [0,infinity[ ?? The answer key says ]0,infinity[

2. ## Re: Range of composite functions

f(x)=sqrt(4-x)
g(x)=x^2

Find the range of gof:
gof(x)=4-x

Shouldn't the range be all real numbers? Because it is a linear function. But the answer key says [0,infinity[ ??
If the range is all real numbers, for which x do we have g(f(x)) = -1? Be sure to calculate g(f(x)) by applying f to x and then g to the result, not 4 - x to x. Concerning "gof(x)=4-x", recall that a function includes not just a way to convert an input into an output, but also a domain. The fact that two functions are equal implies that their domains are the same.

Given two functions:
f(x)=1/x (x cannot be 0)
g(x)1/x^2 (x cannot be 0)

Find the range of fog

fog(x)=x^2

Shouldn't the range by [0,infinity[ ?? The answer key says ]0,infinity[
If the range is [0,infinity[, for which x do we have f(g(x)) = 0? Again, forget fog(x)=x^2 and calculate f(g(x)) by definition for the specific x that you think should work.

3. ## Re: Range of composite functions

Originally Posted by kevinlam2490
Given two functions:

f(x)=sqrt(4-x)
g(x)=x^2

Find the range of gof:
gof(x)=4-x
This is not exactly correct. f(x) exists only if $x\le 4$. Therefore, gof(x)= 4- x with domain $x\le 4$, not the same function as "4- x" (for all real x). Given that x cannot be larger than 4, 4- x cannot be less than 0.

Shouldn't the range be all real numbers? Because it is a linear function. But the answer key says [0,infinity[ ??

Next question:

Given two functions:
f(x)=1/x (x cannot be 0)
g(x)1/x^2 (x cannot be 0)

Find the range of fog

fog(x)=x^2

Shouldn't the range by [0,infinity[ ?? The answer key says ]0,infinity[

But you said "x cannot be 0" so that $x^2$ is never 0 because x cannot be 0. You are not paying enough attention to the domain of the composite function. A "function", even when a formula is given, is not just the formula- it is a specific domain and a formula.
That is, " $f(x)= x^2$" and " $g(x)= x^2$ for x not equal to 0" are different functions!