Animated Solution for Mathematics - Functions: If $f(x)=3x-5$, then $f^{-1}(x)$

Imagine you are standing in front of a magical machine. You feed it a number $x$, and it performs a specific operation: it multiplies that number by $3$ and then subtracts $5$.

Mathematically, we describe this machine as $f(x)=3x-5$.

Now, what if you wanted to run this machine in reverse? What if you had the output and wanted to find the original input? This is the essence of the inverse function, $f^{-1}(x)$. It is the mathematical equivalent of rewinding a movie to see how we got to the current scene.

Before we start calculating, we must ensure our machine is reversible. In the world of JEE mathematics, a function must be bijective—both one-one and onto—to have an inverse.

Think of 'one-one' as a guarantee that every output comes from a unique input; there is no ambiguity. Think of 'onto' as a guarantee that every possible output in our target set is actually reached by the function.

Our function $f(x)=3x-5$ is a simple, elegant straight line. Because its slope is $3$ (which is not zero), it is strictly increasing. It never turns back on itself, so it passes the horizontal line test with flying colors. It is one-one.

Furthermore, as $x$ ranges from $-\infty$ to $+\infty$, the output $y$ also covers the entire set of real numbers $\mathbb{R}$. There are no gaps. It is onto. We have our green light!

Now, let us perform the algebra. We start with the equation:

Our mission is to isolate $x$. Think of this as peeling an onion; we need to remove the layers surrounding $x$ one by one.

First, we address the constant term. We add $5$ to both sides of the equation, which gives us:

The constant is gone, but $x$ is still shackled to the coefficient $3$. To free $x$, we divide the entire equation by $3$. This leaves us with:

We have done it! We have successfully isolated $x$. This expression, $\frac{y+5}{3}$, tells us exactly what the input must have been to produce the output $y$.

There is a profound geometric beauty here that every JEE aspirant should appreciate. If you were to plot $f(x)=3x-5$ and its inverse $f^{-1}(x)=\frac{x+5}{3}$ on the same Cartesian plane, you would notice something striking.

They are perfect mirror images of each other. The mirror is the line $y=x$.

This is not a coincidence; it is a fundamental property of inverse functions. When you swap $x$ and $y$ to find the inverse, you are essentially reflecting the graph across the $45$-degree line.

If you were to fold your graph paper along the line $y=x$, the blue line of the original function would align perfectly with the green line of the inverse. This symmetry is the heartbeat of coordinate geometry.

By mastering this, you are not just solving a problem; you are learning to see the hidden architecture of mathematics. Keep practicing, keep visualizing, and most importantly, keep falling in love with the logic behind the numbers.