Question

Given functions:

f(x)= 1/x-4
g(x)= 4/x+4

Find g(f(x)) and give the restrictions on x.

a) 4/x-4+4, where x not equal to 4
b) 4/x-3+4, where x not equal to 3
c) 4/x-2+4, where x not equal to 2
d) 4/x-1+4, where x not equal to 1

Answer 1

To find g(f(x)), we substitute f(x) into g(x), simplifying the function to (4x-16)/(x+12) with the restrictions that x cannot equal 4 and x cannot equal -12.

Step-by-step explanation:

To find g(f(x)), we need to substitute the function f(x) into g(x). This means wherever we see an x in g(x), we replace it with f(x).

Given that f(x) = 1/(x-4) and g(x) = 4/(x+4), we calculate g(f(x)) = g(1/(x-4)).

We do this by substituting 1/(x-4) into g(x), giving us g(f(x)) = 4/((1/(x-4))+4).

We can combine under a common denominator to get g(f(x)) = 4/((1+4(x-4))/(x-4)).

This simplifies further to g(f(x)) = 4/((x-4+16)/(x-4)) which simplifies to g(f(x)) = 4/((x+12)/(x-4)).

Multiplying by the reciprocal, we get g(f(x)) = 4(x-4)/(x+12). Simplifying this, g(f(x)) = (4x-16)/(x+12).

This simplification indicates that the correct restriction on x is that it cannot equal 4, because if x equals 4, the denominator of f(x) would be zero, which is undefined. Furthermore, x cannot equal -12 because that would make the denominator of the simplified form g(f(x)) equal to zero. Thus the correct restrictions are x ≠ 4 and x ≠ -12.