Given f(x)=1/x; e(x)=x^2 + 4; g(x)=x^3

 

Consider g(e(f(x))), f(e(g(x))), and e(f(g(x))).

 

Question: Make an intelligent conjecture about the relationship between the range of a composite function and the exponential powers of the original functions.

 

g(e(f(x))) and e(f(g(x))) is [math]\frac{1}{x^6}+4[/math].

 

f(e(g(x))) is [math]\frac{1}{x^6+4}[/math].

 

The question is asking to find the relationship between the range of a composite function and the exponential powers of the original functions.

 

For instance: g(e(f(x))), e(f(g(x))), and f(e(g(x))) have an infinity range. The exponentials of the original functions are: -1, 2, and 3.

 

I'm not sure how there is a relationship? I have no idea. Can anyone help me out here?

Can any of the functions ever take negative values?

 

("Have an infinity range" doesn't make any sense as a sentence in the English language; all sentences in mathematics should make sense as grammatical constructs.)