Composition mathIf f(x) = x^2 + 5 and g(x) = sqrt(2x) then find (fog)(x) and (gof)(x) .

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We compose the 2 given functions in this way:

(fog)(x) = f(g(x))

We notice that the variable x was replaced by the function g(x). According to this, we'll write the function f(g(x)) by substituting x by g(x) in the  original expression of f(x):

f(g(x)) = [g(x)]^2 + 5

f(g(x)) = (sqrt 2x)^2 + 5

(fog)(x) = f(g(x)) = 2x + 5

Now, we'll compose gof and we'll get:

(gof)(x) = g(f(x))

We notice that the variable x was replaced by the function f(x). According to this, we'll write the function g(f(x)) by substituting x by f(x) in the  original expression of g(x):

g(f(x)) = sqrt 2f(x)

(gof)(x) = g(f(x)) = sqrt 2(x^2+5)

As we can remark, the result of the 2 compositions is not the same!

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