Animated Solution for Mathematics - Functions: Suppose $f(x)=(x+1{)}^2$ for $x\ge -1$. If $g(x)$ is the function whose graph is the reflection of the graph of $f(x)$ with respect to the line $y=x$, then $g(x)$ equals

A Journey into Symmetry

Imagine you are standing in front of a mirror. You raise your right hand, and your reflection raises its left. This simple, elegant symmetry is the heart of what we are exploring today.

In mathematics, when we talk about the inverse of a function, we are essentially talking about a mirror reflection across the line $y=x$. Let us embark on a journey to understand the function $f(x)=(x+1{)}^2$ and its inverse, $g(x)$.

Visualizing the Parabola

Let us begin by visualizing our given function, $f(x)=(x+1{)}^2$. This is a classic parabola, a shape that appears everywhere in nature, from the path of a projectile to the curve of a satellite dish. Its vertex is at $(-1,0)$.

However, notice the crucial constraint: $x\ge -1$. If we were to draw the entire parabola, it would fail the horizontal line test—meaning for some $y$-values, there would be two different $x$-values. A function must be one-to-one to be invertible.

By restricting our domain to $x\ge -1$, we are selecting only the right-hand branch of the parabola, ensuring that every output $y$ corresponds to exactly one input $x$. This is the secret to making the function perfectly invertible.

The Mirror of $y=x$

Now, imagine drawing the line $y=x$ on your graph. This is the diagonal line that cuts through the origin at a $45$-degree angle. When we reflect our blue curve, $f(x)$, across this line, we obtain the red curve, $g(x)$.

Geometrically, this reflection is the definition of an inverse function. Every point $(a,b)$ on the original curve maps to $(b,a)$ on the reflected curve.

The vertex $(-1,0)$ of our parabola reflects to $(0,-1)$ on our new function. The $y$-intercept $(0,1)$ of the original reflects to $(1,0)$ on the inverse. This symmetry is not just a visual trick; it is the fundamental geometric soul of the inverse function.

The Algebraic Dance

With our visual intuition solid, let us dive into the algebra. We start with the equation:

Our goal is to isolate $x$ to see how the inverse machine works. To undo the square, we take the square root of both sides:

Here is where the domain restriction $x\ge -1$ saves us. Because $x\ge -1$, we know that $x+1\ge 0$. Therefore, the square root of $(x+1{)}^2$ is simply $x+1$, not $\pm (x+1)$.

We have successfully simplified our equation to $\sqrt{y}=x+1$. Now, a simple subtraction gives us:

The Final Reveal

We have solved for $x$ in terms of $y$. To express this as a function of $x$, we swap the variables, giving us $g(x)=\sqrt{x}-1$.

Finally, we must consider the domain. The domain of the inverse function $g(x)$ is the range of the original function $f(x)$. Since $f(x)=(x+1{)}^2$ and $x\ge -1$, the smallest value $f(x)$ can take is $0$.

Thus, the range of $f(x)$ is $f(x)\ge 0$. Consequently, the domain of our inverse function $g(x)$ is $x\ge 0$.

Our final result, $g(x)=\sqrt{x}-1$ for $x\ge 0$, is a beautiful, elegant conclusion to our journey. You have successfully navigated the geometry, the algebra, and the logic of inverse functions.