Animated Solution for Mathematics - Functions: The graph of the function $y=f(x)$ is symmetrical about the line $x=2$, then

Understanding Symmetry

Imagine you are standing in a vast, perfectly still hall. In front of you, there is a mirror placed exactly at the vertical line $x=2$.

Everything to the right of this mirror is reflected perfectly on the left. This is the essence of symmetry in mathematics.

When we say a function $y=f(x)$ is symmetric about the line $x=2$, we are describing a geometric harmony where the function's behavior on one side of the line is a carbon copy of the other.

To translate this visual intuition into the language of algebra, let's start at our mirror line, $x=2$. If we decide to take a walk, moving a distance of $x$ units to the right, we find ourselves at the coordinate $2+x$.

The height of our function at this point is simply $f(2+x)$.

Now, let's return to the mirror line and walk the exact same distance, $x$ units, but this time to the left. We land at the coordinate $2-x$.

The height of our function here is $f(2-x)$.

Here is the core of the problem: because the graph is perfectly symmetric about the line $x=2$, the height of the curve at any point to the right must be identical to the height at the corresponding point to the left.

Mathematically, this means the $y$-values must be equal:

This equation is the algebraic signature of symmetry. It tells us that no matter how far you travel from the line $x=2$, the function maintains the same value at both equidistant points.

This concept is a cornerstone of coordinate geometry and calculus in the JEE syllabus. Whenever you encounter a problem stating that a function is symmetric about a vertical line $x=a$, you can immediately write down the condition:

In our specific problem, $a=2$. Substituting this into our general rule, we get $f(2+x)=f(2-x)$, which matches our second option perfectly.

Remember, symmetry isn't just a property; it's a tool. It allows us to simplify complex integrals, solve functional equations, and visualize graphs without ever needing to plot a single point.

Keep this mirror analogy in your toolkit, and you will find that even the most intimidating functional equations become clear and manageable.