Using g(x) = x + 3/5-x

Question

1. (a) x = g−1(y) =
2. (b) Verify that g(g−1(y)) = y.
3. (c) Verify that g−1(g(x)) = x.

Answer 1

y = (x+3)/(5-x) <--- PLEASE USE PARENTHESIS around the

numerator and hte denominator....

otherwise the function is just 3/5 !!!!

swaps x and y:

x = (y+3)/(5-y)

x(5-y) = y + 3

x(5-y) - y = 3

5x - xy - y = 3

-xy - y = 3 - 5x

xy + y = 5x - 3

y(x+1) = 5x-3

y = (5x-3)/(x+1)

g comp g_inv:

[ (5x-3)/(x+1) + 3 ] divided by [ 5 - (5x-3)/(x+1)]

Multiplies this complex fraction by (x+1)/(x+1) = 1

[(5x-3) + 3(x+1) ] / [5(x+1) - (5x-3)] =

(8x)/ ( 5x + 5 - 5x + 3) =

(8x) / 8 =

x

g_inv comp g:

[(5x-3)/(x+1) + 3] divided by [ 5 - (5x-3)/(x+1)]=

[(5x-3)+ 3(x+1)]/(x+1) divided by [ 5(x+1) - (5x-3)]/(x+1)=

(5x - 3 + 3x + 3)/(x+1) divided by ( 5x+5 - 5x + 3)/(x+1)=

(8x)/(x+1) divided by 8/(x+1) =

(8x)/(x+1) * (x+1)/8=

8x/8 = x