f(x) = x^2 + 5

g(x) = sqrt(2x)

First we will find fog(x)

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

Then we will subsitute x with g(x) in f(x):

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

Now sibsitute with g(x) = sqrt(2x)

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

= 2x + 5

==> **fog(x) = 2x+5**

** **

Now gof(x):

gof(x) = g(f(x)

= sqrt2f(x)

=sqrt[2*(x^2+5)]

= sqrt(2x^2 + 10)

**==> gof(x) = sqrt(2x^2 + 10)**

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!

f(x) = x^2+5.

g(x) = surt(2x).

To find (fog)(x) and (gof)(x).

Solutution:

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

Therefore , put g(x)= sqrt(2x) in place of x in x^2+5.

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

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

**f(g(x)) = 2x+5.. Or (fog) (x) = 2x+5.**

(ii) To find (gof) (x):

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

We put f(x) = x^2+5 in place of x in sqrt(2x).

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

g(f(x)) = sqrt(2x^2+10).

Therefore**(gof)(x) = g(f(x)) = sqrt(2x^2+10).**