Question 1073529
That 2x+3 is the PRODUCT of the functions f and g.


{{{fg(x)=f(x)g(x)=f(x)*(4/x)=2x+3}}}


{{{f(x)*(4/x)=2x+3}}}-------solve this for f(x).



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You want really the composition {{{f(g(x))=2x+3}}}.
(clearer solution method)


Given {{{g(x)=4/x}}}  and the FUNCTION COMPOSITION {{{f(g(x))=2x+3}}}, find the function f(x).


g(x) is used as input into f(x) and the expression for the composition is shown, 2x+3.
You are given g(x), so substitute for its expression.


{{{f(4/x)=2x+3}}}


The input indicated on the left side is {{{4/x}}}, but you want to have an input of x, because you want to find f(x).
What number, r, can you put into 4/x so that the result becomes x?
.
{{{4/r=x}}}


{{{(4/r)*r=x*r}}}


{{{4=r*x}}}


{{{rx=4}}}


{{{rx(1/x)=4(1/x)}}}


{{{highlight_green(r=4/x)}}}


Now you want to replace x in the function {{{f(4/x)=2x+3}}} with {{{r=4/x}}}. 
These must be done to both sides of the function equation.
.
{{{f(4/r)=2r+3}}}


{{{f(4/(4/x))=2(4/x)+3}}}


{{{f((4/4)x)=8/x+3}}}


{{{highlight(f(x)=8/x+3)}}}, 
The function f(x) is now found.