Animated Solution for Mathematics - Inverse Trigonometric Functions: The range of $f(x)=4{\sin }^{-1}\left(\frac{x^2}{x^2+1}\right)$ is

A Journey into Range Analysis

Welcome, future engineers! Today, we are going to unravel the mystery of a composite function. We are tasked with finding the range of

When you first look at this, it might seem like a daunting mountain to climb. However, by using the 'Inside-Out' strategy, we can simplify the path to the solution significantly.

The Inner Core

Let us focus on the inner expression, $g(x)=\frac{x^2}{x^2+1}$. Our goal is to find the range of this inner function first.

Instead of diving into complex calculus, let us use algebraic intuition. We can rewrite $g(x)$ by adding and subtracting $1$ in the numerator:

This allows us to split the fraction into two parts:

This form is much easier to analyze because it isolates the variable $x$ in the denominator.

Building the Bounds

Now, let us build the range of $g(x)$ from the ground up. We know that for any real number $x$, the square is non-negative: $x^2\ge 0$.

If we add $1$ to both sides, we get $x^2+1\ge 1$. As $x^2+1$ grows from $1$ to infinity, its reciprocal $\frac{1}{x^2+1}$ shrinks from $1$ down toward $0$.

Therefore, we have the following inequality:

Applying the negative sign reverses the inequality:

Finally, adding $1$ to all parts gives us:

This simplifies to $0\le g(x)<1$. We have successfully trapped our inner function between $0$ and $1$.

The Final Transformation

Now, we bring back the outer function, the sine inverse. Since ${\sin }^{-1}(u)$ is a strictly increasing function, applying it to our inequality preserves the direction of the signs:

We know that ${\sin }^{-1}(0)=0$ and ${\sin }^{-1}(1)=\frac{\pi }{2}$. So, our inequality becomes:

The final step is to multiply by the constant factor $4$. Multiplying the entire inequality by $4$, we get:

This simplifies to $0\le f(x)<2\pi$.

The range of our function is $