If f(x) = x^2+3      g(x) =sqrt(x)  determine  (fog)(x) and (gof)(x).

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

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)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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!

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

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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