Problem 17 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The graph of the function \(y=f(x)\) is symmetrical about the line \(x=2\), then \(\quad\) [2004] (a) \(f(x)=-f(-x)\) (b) \(f(2+x)=f(2-x)\) (c) \(f(x)=f(-x)\) (d) \(f(x+2)=f(x-2)\)

Step-by-Step Solution

Answer

The correct answer is (b) \(f(2+x)=f(2-x)\).

Step 1: Analyze Symmetry Definition

Given that the function \(y=f(x)\) is symmetrical about the line \(x=2\), we need to understand that this means every point on the graph has a corresponding point on the opposite side of \(x=2\). This symmetry in terms of function values implies that if a point \((a, f(a))\) is on the graph, then \((4-a, f(a))\) will also be on the graph.

Step 2: Set Up the Symmetry Equation

Since the graph is symmetrical about the line \(x=2\), we can express this symmetry mathematically as \(f(2+x) = f(2-x)\). This equation shows that the function values on either side of \(x=2\) are equal at the same distance from the line.

Step 3: Identify the Correct Option

Compare the equation \(f(2+x) = f(2-x)\) with the given options. Option (b) matches exactly with this equation, since it describes the symmetry about \(x=2\) in functional terms.

Step 4: Confirm the Answer

Double-check the conditions provided by the symmetry and verify that none of the other options match the condition \(f(2+x) = f(2-x)\). Options (a), (c), and (d) describe different types of symmetry or transformations, confirming that option (b) is the only correct answer.

Key Concepts

Line Symmetry

In mathematics, line symmetry refers to a situation where a graph is mirrored and perfectly divides itself along a specific vertical or horizontal line. This line is known as the axis of symmetry. In the case of the function presented above, the axis of symmetry is at the line \(x=2\). Whenever a function is said to have line symmetry about a vertical line, like \(x=2\), it means the left and right sides of the graph are mirror images of each other.

For example, if we consider a point \((a, f(a))\) on the graph, then there is a corresponding point \((4-a, f(a))\) reflecting across the line \(x=2\). This reflection concept highlights how every point equidistant from the axis of symmetry provides this mirror effect. To test line symmetry, you can adjust coordinates by a certain amount from the axis, and the function values must remain consistent.

Graphical Symmetry

Graphical symmetry involves noticing visual patterns such that halves of a graph mirror each other. When looking at a graph, if we can fold it along the axis of symmetry and both sides align perfectly, the graph is symmetrical. Seen graphically, this property provides an intuitive understanding that the graph mirrors itself across a line.

For the function \(y=f(x)\), graphical symmetry about the line \(x=2\) suggests that this symmetry line can act as a folding line. The symmetrical nature on the graph confirms that for each point \((x, y)\) located on one side, a matching point exists at \((4-x, y)\) on the other side. When drawing or observing graphs, symmetry simplifies the visualization process and often guides predictions of points and behavior on unknown sections of the function.

Mathematical Symmetry

Understanding mathematical symmetry incorporates precise equivalence in the formulation of functions about symmetry lines. In our situation, the mathematical expression for the symmetrical property of the function is \(f(2+x) = f(2-x)\). This expression mathematically confirms the notion that for each distance \(x\) from the line \(x=2\), the function evaluates to the same outcoming value on both sides.

This method of describing symmetry through an equation allows for more than graphical visualizations. It enables calculations and predictions of unknown values by using known equations reflecting symmetrical properties. Mathematical symmetry leads to insights such as predicting roots, understanding function continuity, and simplifying complex equations by using inherent symmetry properties.