For the functions f(x) = x^2 and g(x) = (x-7), are f (g(x)) = g (f(x)). Also, what is g (f (2))?  

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f(x) = x^2

g(x) = (x-7), 

We need to determine is  f (g(x)) = g (f(x)).

First we will determine f(g(x))

fog (x) = f(g(x)

            = f( x-7)

             = (x-7)^2

              = x^2 - 14x + 49

==> f(g(x)) = x^2 - 14x + 49

Now we will determine...

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f(x) = x^2

g(x) = (x-7), 

We need to determine is  f (g(x)) = g (f(x)).

First we will determine f(g(x))

fog (x) = f(g(x)

            = f( x-7)

             = (x-7)^2

              = x^2 - 14x + 49

==> f(g(x)) = x^2 - 14x + 49

Now we will determine g(f(x):

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

            = g(x^2 )

             = x^2 - 7

==> g(f(x)= x^2 - 7

Then we notice that f(g(x)) does not equal g(f(x))

Now we will calculate g(f(2)):

g(f(x) = x^2 - 7

g(f(2)) = 2^2 - 7

              = 4-7 = -3

==> g(f(2)) = -3

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We are given that f(x) = x^2 and g(x) = (x-7).

Now, to determine g (f(x)), we find the value of f(x) and substitute it in g(x). f(x) = x^2. substituting this in g(x) we get g (f(x)) = x^2 -7.

To determine f (g(x)), we need to find g(x) and substitute the value in f(x). g(x) = (x-7), substituting this in f(x), we get g (f(x)) = (x-7) ^2 = x^2 - 14x + 49.

Therefore we see that g (f(x)) = x^2 -7 and f (g(x)) = x^2 - 14x + 49, which are not the same.

Also, g (f (2)) = 2^2 - 7 = 4 – 7 = -3.

Approved by eNotes Editorial Team