Domain

Now, remember that we can only take the square root of positive numbers. Therefore, the domain of this function is the set of values of $x$ that satisfies this:

$\log_{10} \left( \dfrac{5x-x^2}{4} \right) \geq 0$

Notice that for $\log_{10} p$ :

$\begin{cases} 0<p<1 & \quad \log_{10} p <0\\ p=1& \quad \log_{10} p =0\\ p>1 & \quad \log_{10} p > 0 \end{cases}$

Therefore, to satisfy the inequality above,

$\dfrac{5x-x^2}{4} \geq 1$

$5x-x^2 \geq 4$

$x^2 - 5x + 4 \leq 0$

$(x-1)(x-4) \leq 0$

Draw a number line to solve this, and you should be able to find that the range of values of $x$ is:

$[1,4]$ or $1 \leq x \leq 4$ , whichever way you prefer to write it.

This range satisfies the inequality above, thus it is the domain of the function $y$

$D_y = [1,4] = 1 \leq x \leq 4$

Therefore, $a=1$ , $b=4$ and $a+b = 1+4 = \boxed{5}$