Question 1073536
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I am looking in your post and CAN NOT understand, what is your fg(x) = 2x +3 ?


Is it fog(x)       (composition) ?


Or (fg)(x)        (the product of the two functions) ?


Could you clarify ?



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<U>Comment from student</U>: composition.
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<U>My responce</U>.  OK, composition.


<pre>
1.  First of all, it means that your original formulation in the post was wrong.

    The correct writing is  fog(x)=2x+3.

    The composition is always denoted as fog.


2.  Then you can write fog(x) = f(g(x)) = {{{f(4/x)}}} = 2x + 3.

    Replace here {{{4/x}}} by other variable, for example, t.

    You will get f(t) = 2x + 3.

    But since t = {{{4/x}}}, you have x = {{{4/t}}}, therefore

    f(t) = {{{2*(4/t) + 3}}} = {{{8/t + 3}}}.


    Now you can replace t in both sides by x (it is simply another symbol for the variable), and you will get f(x) = {{{8/x + 3}}}.
</pre>

It is your <U>answer</U>: f(x) = {{{8/x + 3}}}.


<U>Check</U>.  fog(x) = f(g(x) = {{{f(4/x)}}} = {{{(8/((4/x))) + 3}}} = {{{(8x)/4+3}}} = 2x + 3.   Correct !



Solved.