If f(x) = x^2+3 g(x) =sqrt(x) determine (fog)(x) and (gof)(x).
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f(x) = x^2 + 3
g(x) = sqrtx
fog(x) = f(g(x)
= f(sqrtx)
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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 + 3
f(g(x)) = (sqrt x)^2 + 3
(fog)(x) = f(g(x)) = x + 3
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 f(x)
(gof)(x) = g(f(x)) = sqrt (x^2+3)
As we can remark, the result of the 2 compositions is not the same!
f(x) = x^2+3.
g(x) = sqrt(x).
To find (fog)(x) and (gof)(x).
(fog)(x) = f(g(x)) .
We put g(x) = sqrtx in place of x in x^2+3.
f(g(x)) = (sqrt(x))^2 +3 = x+3.
Therefore (fog) (x) = f(g(x)) = x+3.
ii)
To find (gof) (x):
(gof)(x) = g(f(x)).
We put f(x) = x^2+3 in place of x in sqrt(x).
g(f(x)) = sqrt(f(x)+3)
g(f(x)) = sqrt(x^2+3)
Therefore (gof))(x) =sqrt(x^2+3).
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