Question

4. Verify that the functions fand g are inverses of each other by showing that

f(g(x)) = x and g(f(x)) = x.

F(x)=2x-3/x+4
g(x) = 3+4x/2-x

Answer 1

Answer and step-by-step explanation:

We just have to apply one function in the other one. See:

→ f(g(x)):

`f(x) = (2x-3)/(x+4)\\\\f(g(x)) = (2g(x)-3)/(g(x)+4) = (2\cdot(3+4x)/(2-x)-3)/((3+4x)/(2-x)+4)\\\\f(g(x)) = ((6+8x)/(2-x)-3)/((3+4x)/(2-x)+4)\\\\f(g(x)) = (((6+8x)-3(2-x))/(2-x))/(\,\,\,((3+4x)+4(2-x))/(2-x)\,\,\,)=((6+8x)-3(2-x))/((3+4x)+4(2-x))\\\\f(g(x)) =(6+8x-6+3x)/(3+4x+8-4x)=(11x)/(11)\\\\\\\boxed{f(g(x)) = x}`

→ g(f(x)):

`g(x) = (3+4x)/(2-x)\\\\g(f(x)) = (3+4f(x))/(2-f(x)) = (3+4(2x-3)/(x+4))/(2-(2x-3)/(x+4))\\\\g(f(x)) = (3+(8x-12)/(x+4))/(2-(2x-3)/(x+4))\\\\g(f(x)) = ((3(x+4)+(8x-12))/(x+4))/(\,\,\,(2(x+4)-(2x-3))/(x+4)\,\,\,)=(3(x+4)+(8x-12))/(2(x+4)-(2x-3))\\\\g(f(x)) =(3x+12+8x-12)/(2x+8-2x+3)=(11x)/(11)\\\\\\\boxed{g(f(x)) = x}`

What verifies that f and g are inverses of each other.

Q.E.D.