Evaluate the composite functions. Using `f(x)= 1/(2x-7)`  and  `g(x)= x+6` Find  a)  f(g(x))         b)  (f o f)(x)

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lemjay | High School Teacher | (Level 3) Senior Educator

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A. `f(g(x))= ?`
Since f comes before g, to determine f(g(x)), we start with the function f(x).

`f(x) = 1/(2x-7)`

Then, replace the x with g(x).

`f(g(x))=1/(2(g(x))-7)`

Then, plug-in g(x)=x+6.

`f(g(x))=1/(2(x+6)-7)`

`f(g(x))=1/(2x+12-7)`

`f(g(x))=1/(2x+5)`

Hence, `f(g(x))=1/(2x+5) ` .

B. `(fof)(x)=?`

Since the letter that comes first is f, let's start with the function f(x).

`f(x) = 1/(2x-7)`

Since the letter after f is f too, replace the x in the function with f(x).

`f(f(x))=1/(2(f(x))-7)`

Then, plug-in f(x)=1/(2x-7).

`f(f(x))=1/(2(1/2x-7)-7)`

And, simplify.

`f(f(x))=1/(2/(2x-7)-7)`

`f(f(x))= 1/(2/(2x-7)-7) * (2x-7)/(2x-7)`

`f(f(x))=(2x-7)/(2-7(2x-7))`

`f(f(x))=(2x-7)/(2-14x+49)`

`f(f(x))=(2x-7)/(51-14x)`

Hence, `(fof)(x)=(2x-7)/(51-14x)` .

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