Question

If (fog)(x)=7x-1/3 and f(x)=3x+5,find g(x),where g(x)is linear​

Answer 1

Answer: $g(x) = \dfrac73x-\dfrac{16}{9}$

Step-by-step explanation:

Given: $(fog)(x)=7x-\dfrac13$

$f(x)=3x+5$

and g(x) is linear.

A linear function is of the form : $y= mx+c$

Let $g(x)= mx+c$

Then, $(fog)(x)=f(g(x))= f(mx+c)$

$= 3(mx+c)+5\\\\=3mx+3c+5$

Comparing it with the original (fog)(x), we get

$3mx+3c+5=7x-\dfrac{1}{3}$

Comparing coefficient of x and constants separately

$3m=7,\ \ \ \ 3c+5=-\dfrac13\\\\\Rightarrow\ m=\dfrac{7}{3},\ \ \ \ 3c =-\dfrac{1}{3}-5\\\\\Rightarrow\ m=\dfrac{7}{3},\ \ \ \ 3c =-\dfrac{16}{3}\\\\\Rightarrow\ m=\dfrac{7}{3},\ \ \ \ c =-\dfrac{16}{9}$

So, $g(x) = \dfrac73x-\dfrac{16}{9}$