Animated Solution for Mathematics - Functions: Let the domain of the function $f(x)={\log }_3{\log }_5(7-{\log }_2(x^2-10x+85))+{\sin }^{-1}\left(\left|\frac{3x-7}{17-x}\right|\right)$ be $(\alpha ,\beta ]$. Then $\alpha +\beta$ is equal to :

Welcome, future IITians! Today, we are going to embark on a journey through a function that looks like a tangled web of nested logarithms and inverse trigonometric functions.

The function is defined as:

To find the domain, we must ensure that every part of this function is mathematically sound. We have two primary constraints: the logarithmic term and the inverse sine term.

The outermost logarithm is ${\log }_3(\cdot )$. For this to be defined, its argument must be strictly positive.

We set the following inequality:

Using the property that ${\log }_a(b)>c⟺b>a^c$ (for $a>1$), we obtain:

Rearranging this, we find:

Applying the exponential property again, we get:

This leads to the quadratic inequality:

Factoring this, we get $(x-3)(x-7)<0$. The solution to this is the open interval $x\in (3,7)$.

Now, let's turn our attention to the second part of our function: ${\sin }^{-1}\left(\left|\frac{3x-7}{17-x}\right|\right)$. The domain of ${\sin }^{-1}(u)$ is $u\in [-1,1]$.

Thus, we must satisfy:

Since both sides are non-negative, we can square them:

Expanding these squares, we get:

Bringing everything to one side, we arrive at:

Dividing by $8$, we get $x^2-x-30\le 0$. Factoring this, we have $(x-6)(x+5)\le 0$, which yields the interval $x\in [-5,6]$.

We have our two domains: $(3,7)$ and $[-5,6]$. The domain of the entire function $f(x)$ is the intersection of these two sets.

Looking at the number line, the values that satisfy both conditions are those in the interval $(3,6]$.

The problem states the domain is $(\alpha ,\beta ]$. By comparing $(3,6]$ with $(\alpha ,\beta ]$, we identify $\alpha =3$ and $\beta =6$.

The final sum is:

We have conquered the labyrinth! Keep practicing, and remember: every complex problem is just a series of simple steps.