Same Domain

Given that $X=\{-6,6\}$ is the domain of the two functions $f$ and $g$ where $f$ and $g$ are defined as follows :--->

$f(x)=ax+b\quad \text{and} \quad g(x)=-x^2+3x-1$

Now, let us use the values from the domain $X$ to find the values of $f(6),f(-6),g(-6)\text{ and }g(6).$

$f(6)=6a+b \quad \quad, \quad g(6)=(-(6)^2+3(6)-1)=(-36+18-1)=(-19)$

$f(-6)=-6a+b \quad ,\quad g(-6)=(-(-6)^2+3(-6)-1)=(-36-18-1)=(-55)$

Since $f=g$ , so $f(6)=g(6)$ and $f(-6)=g(-6)$ . So, we get the two following equations --->

$6a+b=(-19)...(i) \quad \text{ and }\quad b-6a=(-55)....(ii)$

Adding eq. (i) and (ii), we get --->

$6a+b+b-6a=(-19)+(-55) \implies 2b=(-74) \implies b=\boxed{(-37)}$

Subtracting eq. (ii) from (i), we get ---->

$6a+b-b+6a=(-19)-(-55) \implies 12a=36 \implies a=\boxed{3}$

So, the required value of $(a-b)=(3-(-37))=3+37=\boxed{40}$

-6a+b=-36-18-1 6a+b=-36+18-1 a=3 b=-37