Question

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

Answer:
`g(x) = \frac73x-(16)/(9)`

Explanation:

Given:
`(fog)(x)=7x-\frac13`

`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-(1)/(3)`

Comparing coefficient of x and constants separately

`3m=7,\ \ \ \ 3c+5=-\frac13\\\\\Rightarrow\ m=(7)/(3),\ \ \ \ 3c =-(1)/(3)-5\\\\\Rightarrow\ m=(7)/(3),\ \ \ \ 3c =-(16)/(3)\\\\\Rightarrow\ m=(7)/(3),\ \ \ \ c =-(16)/(9)`

So,
`g(x) = \frac73x-(16)/(9)`