Animated Solution for Mathematics - Inverse Trigonometric Functions: If the domain of the function $f(x)={\sec }^{-1}\left(\frac{2x}{5x+3}\right)$ is $[\alpha ,\beta )\cup (\gamma ,\delta ]$, then $|3\alpha +10(\beta +\gamma )+21\delta |$ is equal to

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving an equation; we are exploring the boundaries of existence for a function.

When we look at $f(x)={\sec }^{-1}\left(\frac{2x}{5x+3}\right)$, we aren't just looking at symbols on a page. We are looking at a gatekeeper.

The inverse secant function is notoriously picky. It refuses to accept values that are 'too small' or 'too close to zero.' Specifically, for any input $u$, the function ${\sec }^{-1}(u)$ is only defined if $|u|\ge 1$.

This means our argument, $\frac{2x}{5x+3}$, must live in the wild, outside the safe, forbidden zone of $(-1,1)$.

The First Front ($u\le -1$)

Let us confront the first condition: $\frac{2x}{5x+3}\le -1$. Now, I want you to pause. Your instinct might be to multiply both sides by $(5x+3)$.

Resist that urge! In the world of inequalities, cross-multiplication is a trap. If $(5x+3)$ is positive, the inequality stays the same; if it is negative, the inequality flips.

Since we don't know the sign of $x$, we cannot risk it. Instead, we bring the $-1$ to the left side:

By finding the common denominator, we transform this into a single, elegant fraction:

Now, we enter the realm of the Wavy Curve Method. Our critical points are the roots of the numerator ($x=-\frac{3}{7}$) and the roots of the denominator ($x=-\frac{3}{5}$).

Plotting these on the number line, we see three regions. Testing the signs, we find that the expression is negative between our critical points.

Thus, for this case, $x\in \left(-\frac{3}{5},-\frac{3}{7}\right]$. Notice the bracket at $-\frac{3}{7}$ is closed because the inequality allows equality, but the bracket at $-\frac{3}{5}$ is open because the denominator cannot be zero.

The Second Front ($u\ge 1$)

We are not done yet. We must now conquer the second condition: $\frac{2x}{5x+3}\ge 1$. Again, we move the $1$ to the left side to avoid the cross-multiplication trap:

Simplifying this, we get:

To make this cleaner, let us divide the entire inequality by $-3$. Remember the golden rule: dividing by a negative number flips the inequality sign! We are left with:

Our critical points here are $x=-1$ and $x=-\frac{3}{5}$. Applying the Wavy Curve Method again, we find the interval where the expression is less than or equal to zero. This gives us $x\in \left[-1,-\frac{3}{5}\right)$.

The Synthesis

Now, we bring our two findings together. The domain of our function is the union of these two sets:

Comparing this to the given form $[\alpha ,\beta )\cup (\gamma ,\delta ]$, we can identify our constants with absolute clarity:

We have reached the summit. The problem asks us to evaluate the expression $|3\alpha +10(\beta +\gamma )+21\delta |$. Let us substitute our values with care:

Calculating step-by-step:

And there it is. The answer is 24. It is not just a number; it is the result of your patience, your adherence to the rules of inequalities, and your ability to visualize the domain.