Use composition to verify the functions are inverses fx x22

Use composition to verify the functions are inverses. f(x) = (x-2)^2, x > 2; g(x) = squareroot x + 2, x > 0 f(x) = 5x^3 + 1 g(x) = (x - 1/5)^1/3

Solution

1) compose the functions that is, plug x into one function, plug that function into the inverse function, and then simplify) and verify that you end up with just \"x\".

f(x) = ( x-2)^2

g(x) = sqrtx +2

(f o g)(x) = ( sqrtx + -2)^2 = (sqrtx)^2 = x

(g o f)(x) = sqrt(x-2)^2 +2 = (x -2) + 2 = x

f (x) and g(x) are inverses of each other.

2)   f(x) = 5x^3 +1

g(x) = {( x-1)/5}^1/3

compose the functions that is, plug x into one function, plug that function into the inverse function, and then simplify) and verify that you end up with just \"x\".

(f o g)(x) = 5({(x-1)/5}^1/3)^3 +1

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

(g o f)(x) ={ ( 5x^3 -1 +1)/5}^1/3

= (5x^3/5)^1/3

= x

f (x) and g(x) are inverses of each other.