How do I find a function g that yields a specific answer when composed with f?

Question

Let $f(x)=\frac x{x-2}$ . Find a function $y=g(x)$ so that $(f\circ g)(x)=4x$.

The solution is $g(x)=\frac{8x}{4x-1}$, but I don't know how to arrive at this answer. Please help me to find the steps to solve this problem.

score 8 · Answer 1 · answered Sep 10 '16 at 16:34

8

If $g(x)=y$ then $$4x=f(g(x))=f(y)=\frac{y}{y-2}$$ hence $4xy-8x=y$, from which you get $$y=\frac{8x}{4x-1}$$

Therefore $g(x)=y=\frac{8x}{4x-1}$.

answered Sep 10 '16 at 16:34

Angelo Rendina

1,893

Can show how $4xy-8x=y$ leads to $y=\frac{8x}{4x-1}$? I'm stuck there. – error Sep 10 '16 at 16:55
5

@error Solve for $y$. "Move" all the $y$ on a side, and everything else on the other. So $4xy-8x=y$ becomes (moving $y$ to the left) $4xy-y=8x$. Then factorize $y(4x-1)=8x$ and then divide both terms. This is the standard procedure in linear equations in one variable! – Angelo Rendina Sep 10 '16 at 16:58
I don't see why adding $y$ makes it any easier. It looks more complicated and less intuitive to me. – user541686 Sep 11 '16 at 06:44
@Mehrdad it does not make any difference, in theory. But, especially for beginners, $y$ looks more like a "number" while $g(x)$ is less transparent as it reminds of a function (although, for fixed $x$, it is of course a number). – Angelo Rendina Sep 11 '16 at 11:44

score 6 · Answer 2 · edited Sep 10 '16 at 21:18

6

Just write $(f\circ g)(x)$, which is $$\dfrac{g(x)}{g(x)-2}.$$ Now solve the equation $$\dfrac{g(x)}{g(x)-2}=4x$$ with respect to the variable $g(x)$.

edited Sep 10 '16 at 21:18

Carsten S

8,726

answered Sep 10 '16 at 16:35

score 2 · Answer 3 · answered Sep 10 '16 at 16:35

2

$$f(x)=\frac { x }{ x-2 } ,y=g(x),(f\circ g)(x)=4x\\ \left( f\circ g \right) (x)=f\left( g\left( x \right) \right) =\frac { g\left( x \right) }{ g\left( x \right)-2 } =4x\Rightarrow \quad g\left( x \right)=4xg\left( x \right)-8x\Rightarrow g\left( x \right)\left( 4x-1 \right) =8x\\ \Rightarrow g\left( x \right)=\frac { 8x }{ 4x-1 } \\ $$

answered Sep 10 '16 at 16:35

haqnatural

21,578

I'm getting [Math Processing Error] on this answer. – Sep 10 '16 at 20:06
can you show a reason? – haqnatural Sep 10 '16 at 20:16
Unfortunately, that's all MathJax tells me. – Sep 10 '16 at 20:43

score 1 · Accepted Answer · answered Sep 10 '16 at 16:55

Ok. We know that $f(x)=\frac{x}{x-2}$

we also know that if we replace the variable x in f(x) with g(x) (let's call it "G" for simplicity) we must get 4x as a result. Let's write that down: $$\frac{G}{G-2}=4x$$ $$G=4x(G-2)$$ $$G=4xG-8x$$ $$G-4xG=-8x$$ $$G(1-4x)=-8x$$ $$G=\frac{-8x}{(1-4x)}$$ $$G=\frac{8x}{4x-1}$$

How do I find a function g that yields a specific answer when composed with f?

4 Answers4