Determining the Composition of Functions: Simplifying the Expression g(f(x)) = x^(-6)

Find the composition of the function.f(x)=1/x^2 and g(x)=1/x^3 . Find g(f(x)).

To find the composition of two functions, we need to substitute the expression for one function into the other

To find the composition of two functions, we need to substitute the expression for one function into the other.

Given f(x) = 1/x^2 and g(x) = 1/x^3, we want to find the composition g(f(x)).

To find g(f(x)), we substitute f(x) into g(x).

So, g(f(x)) = g(1/x^2).

Now, substitute 1/x^2 into g(x), we get:

g(f(x)) = g(1/x^2) = 1/(1/x^2)^3

To simplify this expression, we need to apply the rules of exponents.

Recall that when we raise a fraction to a power, we raise both the numerator and the denominator to that power.

So, (1/x^2)^3 = (1^3)/(x^2)^3 = 1/x^6 = x^(-6).

Therefore, g(f(x)) = x^(-6)

In conclusion, g(f(x)) = x^(-6).

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