Answer: [tex]g(x) = \dfrac73x-\dfrac{16}{9}[/tex]
Step-by-step explanation:
Given: [tex](fog)(x)=7x-\dfrac13[/tex]
[tex]f(x)=3x+5[/tex]
and g(x) is linear.
A linear function is of the form : [tex]y= mx+c[/tex]
Let [tex]g(x)= mx+c[/tex]
Then, [tex](fog)(x)=f(g(x))= f(mx+c)[/tex]
[tex]= 3(mx+c)+5\\\\=3mx+3c+5[/tex]
Comparing it with the original (fog)(x), we get
[tex]3mx+3c+5=7x-\dfrac{1}{3}[/tex]
Comparing coefficient of x and constants separately
[tex]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}[/tex]
So, [tex]g(x) = \dfrac73x-\dfrac{16}{9}[/tex]
