Animated Solution for Mathematics - Inverse Trigonometric Functions: The domain of ${\sin }^{-1}[{\log }_3(x/3)]$ is

Peeling Back the Layers

Imagine you are standing before a complex mathematical structure. It looks intimidating, perhaps even a bit alien. Today, we are looking at the function $f(x)={\sin }^{-1}[{\log }_3(\frac{x}{3})]$.

It is a composite function, a nested structure where one operation is tucked inside another. In mathematics, as in life, when you face something complex, the best strategy is to peel it back layer by layer. Let us embark on this journey together.

The Gatekeeper

Every function has its own set of rules, its own 'domain'—the playground where it is allowed to exist. Our outermost function is the inverse sine, ${\sin }^{-1}(u)$.

Think of this as a strict gatekeeper. It only accepts inputs that fall within the closed interval $[-1,1]$. If you try to pass it a value like $1.5$ or $-2$, the gatekeeper shuts the door; the function simply does not exist there.

So, our first step is to acknowledge this boundary: the entire expression inside the inverse sine, which we will call $u$, must satisfy $-1\le u\le 1$. This is our anchor.

The Logarithmic Bridge

Now, we look at what is inside that gatekeeper: the logarithmic expression ${\log }_3(\frac{x}{3})$. This is our bridge to the variable $x$.

We substitute this into our boundary condition, creating a double inequality:

This inequality tells us that the output of our log function is trapped. It cannot be less than $-1$, and it cannot be greater than $1$. We are essentially defining the 'safe zone' for our variable $x$.

The Final Unveiling

Now, we must free $x$ from the clutches of the logarithm. To do this, we use the inverse operation: exponentiation. We raise the base, $3$, to the power of each part of our inequality.

We must ask: is the base greater than $1$? Yes, the base is $3$. Because $3>1$, the logarithmic function is monotonically increasing, meaning it preserves the order of values.

We can safely exponentiate without flipping our inequality signs. This gives us:

Calculating the powers is straightforward: $3^{-1}$ becomes $\frac{1}{3}$, and $3^1$ remains $3$. Our inequality now looks like this:

To isolate $x$ completely, we multiply the entire inequality by $3$. The left side becomes $3\times \frac{1}{3}=1$. The middle term becomes $x$. The right side becomes $3\times 3=9$.

The Elegance of the Result

We have arrived at our destination: $1\le x\le 9$. The domain of our function is the closed interval $[1,9]$.

It is a beautiful, clean result. We started with a terrifying-looking composite function and, by respecting the rules of the gatekeeper and the properties of the logarithmic bridge, we found the exact region where the function lives.

Remember, math is not about memorizing steps; it is about understanding the flow of logic. You have successfully navigated the layers. The final domain is $[1,9]$.