Question

If $f(x)=\frac{3x+2}{4x-1}$ and $g(x)=\frac{x+2}{4x-3}$, prove that (gof) (x) = (fog) (x) = x.

Answer 1

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Answer 2

Step-by-step explanation:

In this question

We have been given that

f(x) = $\frac{3x+2}{4x-1}$

g(x) = $\frac{x+2}{4x-3}$

To prove - (fog)(x) = (fog)(x) = x

Proof -: fog(x) = f(g(x))

                       = f($\frac{x\ +\ 2}{4x\ -\ 3}$)

                       = $\frac{3(\frac{x\ +\ 2}{4x\ -\ 3})\ +\ 2}{4(\frac{x\ +\ 2}{4x\ -\ 3})\ -\ 1}$

Taking L.C.M we get,

= $\frac{3x\ +\ 6\ +\ 8x\ -\ 6}{4x\ +\ 8\ -\ 4x\ +\ 3}$

= $\frac{11x}{11}$

= x

gof(x) = g(f(x))

          = g($\frac{3x\ +\ 2}{4x\ -\ 1}$)

          = $\frac{\frac{3x\ +\ 2}{4x\ -\ 1}\ +\ 2}{4(\frac{3x\ +\ 2}{4x\ -\ 1})\ -\ 3}$

Taking L.C.M we get,

= $\frac{3x\ +\ 2\ +\ 8x\ -\ 2}{12x\ +\ 8\ -\ 12x\ +\ 3}$

= $\frac{11x}{11}$

= x

Therefore fog(x) = gof(x) = x

Hence prooved