F(c) = 1/x+5 -1 find the inverse function of f(x) and it’s domain

f ( x ) = 1/x+5 - 1

y = 1/x+5 - 1

Add both sides 1

y + 1 = 1/x+5 - 1 + 1

y + 1 = 1/x+5

Multiply both sides by ( x + 5 )

( x + 5 ) × ( y + 1 ) = ( x + 5 ) × 1 / x + 5

xy + x + 5y + 5 = 1

x( y + 1 ) + 5y + 5 = 1

Subtract both sides 5

x( y + 1 ) + 5y + 5 - 5 = 1 - 5

x( y + 1 ) + 5y = - 4

Subtract both sides 5y

x( y + 1 ) + 5y - 5y = - 4 - 5y

x( y + 1 ) = - 5y - 4

Divide both sides by ( y + 1 )

x( y + 1 ) ÷ ( y + 1 ) = ( - 5y - 4 ) ÷ ( y + 1 )

x = - 5y - 4 / y + 1

Thus the inverse function of f(x) is :

y = - 5x - 4 / x + 1 ( D : R - { - 1 } )

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