Find the domain

Question

Given f(x) = 5x -3 and g(x) = x ² + 4x, solve (g / f) (x) and the domain of (g/f)(x).

Answer 1

The functions are f(x) = 5x - 3 and g(x) = x ² + 4x.

The domain of f(x) is all real numbers.

The domain of g(x) is all real numbers.

Find (g/f)(x).

(g/f)(x)   =   g(x)/f(x)

              =   (5x - 3) / (x^2 + 4x)

Denominator will not be zero

x^2 + 4x   =   0 

x ( x + 4 )   =   0

x  =  0         ;       x + 4   =   0

x  =  0         ;       x   =   -4

The value of the denominator is zero when x = 0, -4

Hence the domain of the function is All real numbers except x = 0, -4