What is f(g(x)) and its domain if..? ?

Question

What is f(g(x)) and its domain if:

f(x)= x+5 / IxI-5

and

g(x)=2x +7?

Answer 1

any x so that g(x) ╪ 5 or -5...{ -1 , -6 }

Answer 2

Is x + 5 the numerator and |x| - 5 the denominator?

If so, we write (x+5) / (IxI-5)

If 2x + 7 > 0, we substitute this expression and get

f(g(x)) = (g(x) + 5) / (g(x) - 5)

= ( 2x + 7 + 5) / (2x + 7 - 5)

= (2x + 12) / (2x + 2)

= 2(x + 6) / 2(x + 1)

= (x + 6) / (x + 1)

This expression is defined for all x except for

x + 1 = 0

i.e. x = -1

and so -1 is excluded from the domain.

If 2x + 7 < 0,

i.e. 2x < -7

and so x < -7/2,

then |2x + 7| = -2x - 7, and so

f(g(x)) = (2x + 7 + 5) / (-2x -7 -5)

= (2x + 12) / (-2x - 12)

This expression is undefined when 2x + 12 = 0,

i.e. when x = -6

and so -6 is excluded from the domain.

For all other values on the left of -7/2, the expression is equal to -1

Hence the domain of f(g(x)) is

{x: x is not equal to -6 and x is not equal to -1}

and f(g(x)) = -1 when x < -6 or -6 < x < -7/2

................. (x + 6) / (x + 1) when -7/2 ≤ x < -1 or x > -1