# 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)

= (sqrtx)^2 + 3

= x+ 3

**==> fog(x) = x+ 3**

gof(x) = g(f(x)

= g(x^2 + 3)

= sqrt(x^2 + 3)

**==> gof(x) = sqrt(x^2 + 3)**

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).