SOLUTION: Given that g(x)=4/x , x is not equal to 0 and fg(x)=2x+3 find the function f.

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Question 1073529: Given that g(x)=4/x , x is not equal to 0 and fg(x)=2x+3 find the function f.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
That 2x+3 is the PRODUCT of the functions f and g.

fg%28x%29=f%28x%29g%28x%29=f%28x%29%2A%284%2Fx%29=2x%2B3

f%28x%29%2A%284%2Fx%29=2x%2B3-------solve this for f(x).


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You want really the composition f%28g%28x%29%29=2x%2B3.
(clearer solution method)

Given g%28x%29=4%2Fx and the FUNCTION COMPOSITION f%28g%28x%29%29=2x%2B3, 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%284%2Fx%29=2x%2B3

The input indicated on the left side is 4%2Fx, 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%2Fr=x

%284%2Fr%29%2Ar=x%2Ar

4=r%2Ax

rx=4

rx%281%2Fx%29=4%281%2Fx%29

highlight_green%28r=4%2Fx%29

Now you want to replace x in the function f%284%2Fx%29=2x%2B3 with r=4%2Fx.
These must be done to both sides of the function equation.
.
f%284%2Fr%29=2r%2B3

f%284%2F%284%2Fx%29%29=2%284%2Fx%29%2B3

f%28%284%2F4%29x%29=8%2Fx%2B3

highlight%28f%28x%29=8%2Fx%2B3%29,
The function f(x) is now found.