Problem 157 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The graph of the function \(y=f(x)\) is symmetrical about the line \(x=2\), then (A) \(f(x+2)=f(x-2)\) (B) \(f(2+x)=f(2-x)\) (C) \(f(x)=f(-x)\) (D) \(f(x)=-f(-x)\)

Step-by-Step Solution

Answer

Option (B) is correct: \(f(2+x) = f(2-x)\).

Step 1: Identify the Symmetry

The function is symmetric about the line \(x = 2\). This implies that for any point \((x, f(x))\) on the graph, there exists a corresponding point \((4-x, f(4-x))\) that is also on the graph. In other words, points are mirrored across the line \(x = 2\).

Step 2: Transform the Variables

To determine the symmetry property concerning a shift or reflection, it's useful to transform the function according to the line of symmetry. The transformation required here is subtracting 2 from the input to center the symmetry at 0. That is, let \(g(x) = f(x+2)\). Then the symmetry would imply \(g(x) = g(-x)\) based on the centralized axis of symmetry.

Step 3: Restate in Terms of the Original Function

Returning to the original function notation, the relationship \(g(x) = g(-x)\) translates back to the original variable as \(f(x+2) = f(4-x-2) = f(2-x)\), showing that \(f(x+2) = f(2-x)\).

Step 4: Determine the Correct Option

The transformation steps show that the function \(f(x)\) satisfies the equation \(f(2+x) = f(2-x)\). Therefore, the correct property satisfies option (B).

Key Concepts

Graph Symmetry

Graph symmetry is a fascinating concept where parts of a graph are mirror images of each other. When we say a graph is symmetrical, it means if you could fold the graph along a certain line, both halves would match perfectly. In the context of functions, this line is often called the line of symmetry or axis of symmetry.

To visualize this, imagine the function graph of a quadratic equation, like a parabola. If it opens upwards, it might have a vertical line of symmetry through its vertex. This means that if you look at the graph on one side of this line, it will look the same as on the other side.

In mathematical exercises, recognizing the symmetry in a graph allows you to predict and confirm the behavior of a function without needing to compute every single point. It simplifies problems by reducing them to smaller parts, enabling easier predictions and calculations.

Functions and Transformations

Functions and transformations go hand in hand when it comes to understanding how graphs behave or change. Transformations involve shifting, stretching, compressing, or reflecting a graph of a function.

Let's unpack some transformations:

• **Shift:** Moving the entire graph horizontally or vertically. For example, shifting a function right by 2 units involves replacing its variable, say, \(x\), with \(x-2\). This moves each point on the graph to the right by 2 units.
• **Reflection:** Flipping a graph over a line, like the x-axis or the y-axis. In our exercise, reflecting over the line \(x=2\) involves altering the symmetry so that points on the right mirror those on the left.

The beauty of transformations is how they maintain the core characteristics of the function while altering their appearance on the graph. By mastering these, you can predict the effects of different operations on functions easily.

Axis of Symmetry

The axis of symmetry is a vital part of understanding symmetrical functions. It is the imaginary line that divides the graph into two mirror-image halves. For many functions, this axis helps us understand their behavior and basic structure.

Consider a parabola, which typically has a vertical axis of symmetry. This vertical line goes through the vertex of the parabola. For instance, if our function graph looks like \(y = (x-2)^2\), the axis would be the line \(x=2\).

To find the axis of symmetry in a function, observe where the function repeats its values. In shifts and reflections, like in the exercise provided, understanding the axis lets you realign the function to explore and explain its symmetry more clearly. This knowledge is essential in solving equations and graph-related problems efficiently. The axis of symmetry is not only a line but a critical guide for symmetry in function analysis.