Show that f and g are inverse functions algebraically.
f(x)=x^3/2
g(x)=3 cuberoot 2x
My work so far:
(3sqrt2x)^3
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2
I need help on the numeral part of the expression
To show that f and g are inverse functions algebraically, we need to demonstrate that when the compositions of f and g, and g and f are both evaluated, they result in the identity function.
Let's start by finding the composition of f and g:
(f ∘ g)(x) = f(g(x))
Using the given functions f(x) = x^(3/2) and g(x) = 3√(2x), we can substitute g(x) into f as follows:
f(g(x)) = (g(x))^(3/2)
Substituting g(x) = 3√(2x) into the expression:
f(g(x)) = (3√(2x))^(3/2)
To simplify further, we can rewrite (3√(2x))^(3/2) as (2x)^(3/4):
f(g(x)) = (2x)^(3/4)
Next, we need to determine the composition of g and f:
(g ∘ f)(x) = g(f(x))
Using the given functions f(x) = x^(3/2) and g(x) = 3√(2x), we can substitute f(x) into g as follows:
g(f(x)) = 3√(2f(x))
Substituting f(x) = x^(3/2) into the expression:
g(f(x)) = 3√(2(x^(3/2)))
To simplify further, we can rewrite 3√(2(x^(3/2))) as (3√2) * (x^(3/2))^(1/3):
g(f(x)) = (3√2) * (x^(3/2))^(1/3)
Let's now simplify each of the compositions:
(f ∘ g)(x) = (2x)^(3/4)
(g ∘ f)(x) = (3√2) * (x^(3/2))^(1/3)
To demonstrate that f and g are inverse functions, we want to show that the compositions are equal to the identity function, which is x.
(f ∘ g)(x) = x
(g ∘ f)(x) = x
Comparing the two compositions to the identity function, we can conclude that f and g are inverse functions algebraically if:
(2x)^(3/4) = x
(3√2) * (x^(3/2))^(1/3) = x
To solve the equation (2x)^(3/4) = x, we can raise both sides to the fourth power:
[(2x)^(3/4)]^4 = x^4
8x^3 = x^4
Rearranging the terms:
x^4 - 8x^3 = 0
Factoring out an x^3:
x^3(x - 8) = 0
This equation gives two solutions:
x^3 = 0 => x = 0
x - 8 = 0 => x = 8
Both x = 0 and x = 8 satisfy the equation (2x)^(3/4) = x.
For the equation (3√2) * (x^(3/2))^(1/3) = x, we can simplify it as follows:
(3√2) * (x^(3/2))^(1/3) = x
(3√2) * x^(3/2 * 1/3) = x
(3√2) * x^(1/2) = x
Squaring both sides:
[(3√2) * x^(1/2)]^2 = x^2
(3√2)^2 * (x^(1/2))^2 = x^2
18 * x = x^2
Rearranging the terms:
x^2 - 18x = 0
Factoring out an x:
x(x - 18) = 0
This equation gives two solutions:
x = 0
x - 18 = 0 => x = 18
Both x = 0 and x = 18 satisfy the equation (3√2) * (x^(1/2)) = x.
Therefore, we have found four values of x that satisfy both equations. This means that f and g are inverse functions as they undo each other's computations.