if f(x) = 7/x+5 and g(x) = 1/x what is (fog)(x)

This question is what we called composite functions in Mathematics. A composite function is defined as applying one function to the results of another. If we say the result of f() is sent through g(), we can express mathematically as follows: 

`(g@ f)(x)`

Which translates as the following: 

`g(f(x))`

And vice versa, like in our example, we say the result of g() is send through f():

`(f @ g)(x)`

which translates as: 

`f (g(x))`

PLEASE...

This question is what we called composite functions in Mathematics. A composite function is defined as applying one function to the results of another. If we say the result of f() is sent through g(), we can express mathematically as follows: 

`(g@ f)(x)`

Which translates as the following: 

`g(f(x))`

And vice versa, like in our example, we say the result of g() is send through f():

`(f @ g)(x)`

which translates as: 

`f (g(x))`

PLEASE NOTE: `(g @ f(x))`   is NOT equal to `(g @ f) ` 

Now let's answer our question:

`f(x) = 7/(x+5)`

`g(x) =1/x`

Now write what we solving for as shown above: 

`(f@g) (x) = f(g(x))`

Now where ever you see x substitute g(x):

`f(g(x)) = 7/ (g(x) +5)`

We know the value of g(x), now we substitute it 

`f(g(x)) = 7/((1/x) + 5)` 

We can simplify this to get rid of the fraction in the denominator.  Multiply 5 by x/x to get a common denominator and then add them together.

`f(g(x)) = 7/((1/(x)) + ((5x)/x))`

`f(g(x)) = 7/((1 + 5x)/x)`

We can simplify this more by multiplying the numerator and the denominator by x.  This will get rid of the fraction in the denominator.

`f(g(x)) = 7/((5x + 1)/x) * x/x = (7x)/(5x + 1)`

SUMMARY: 

• Convert `(f@g) ` into `f (g(x))`

 

• ANSWER: 


• 
  `(f@g) = (7x)/(5x + 1)`