Problem 65 - A Complete Resource Book in Mathematics for JEE Main

Question

Suppose \(f(x)=(x+1)^{2}\) for \(x \geq-1\). 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) \(-\sqrt{x}-1, x \geq 0\) (B) \(\frac{1}{(x+1)^{2}}, x>-1\) (C) \(\sqrt{x+1}, x \geq-1\) (D) \(\sqrt{x}-1, x \geq 0\)

Step-by-Step Solution

Answer

The function is \(g(x) = \sqrt{x} - 1, x \geq 0\).

Step 1: Understanding the Reflection

To reflect a function's graph with respect to the line \(y=x\), we essentially switch the roles of \(x\) and \(y\) in the function's equation. This gives us a new function \(g(x)\) where \(x\) is the output of the original function and \(y\) becomes the input.

Step 2: Write Original Function as \(y=f(x)\)

Given \(f(x)=(x+1)^{2}\), write it as \(y=(x+1)^2\). This equation represents the original function before reflection.

Step 3: Swap \(x\) and \(y\)

To find \(g(x)\), swap \(x\) and \(y\) from the equation \(y=(x+1)^2\). This gives \(x = (y+1)^2\).

Step 4: Solve for \(y\)

To express \(y\) as a function of \(x\), solve \(x = (y+1)^2\) for \(y\). Take the square root of both sides: \(\sqrt{x} = y + 1\).

Step 5: Simplify \(y\) in terms of \(x\)

Rearrange to find \(y\): \(y = \sqrt{x} - 1\). This equation tells us that \(g(x) = \sqrt{x} - 1\).

Step 6: Define the Domain of \(g(x)\)

Since \(f(x)\) is defined for \(x \geq -1\), the reflection \(g(x) = \sqrt{x} - 1\) will be defined for \(x \geq 0\) to ensure that only real, non-negative outputs from \(f(x)\) are used as inputs.

Key Concepts

Graph Transformation

Graph transformation is a powerful technique that allows us to visualize changes applied to a function. One common transformation is reflection. When a graph is reflected over a line, it flips over that line as if mirrored.
In this exercise, we reflect the function against the line \(y = x\). Reflecting over this line swaps the \(x\) and \(y\) values of each point. Essentially, it changes the function into its inverse. This results in a new graph, \(g(x)\), that looks like it was turned inside out compared to the original \(f(x)\).
When solving such a problem:

• First, write the original function as \(y = f(x)\).
• Next, switch \(x\) and \(y\) in the equation.
• The result is the transformed function, showing a reflection or inverse based on symmetry.

Understanding how graph transformations work helps in visualizing how functions behave when altered.

Inverse Functions

Inverse functions are a way to "undo" a function. If you have a function \(f(x)\), its inverse, \(g(x)\), will return the original \(x\) values when applied to the \(y\) values of \(f(x)\).
To find the inverse of a function, swap \(x\) and \(y\) in the equation, then solve for \(y\) in terms of \(x\). The inverse function will represent all input-output pairs reversed.
For our exercise:

• Given \(f(x) = (x+1)^2\), we first wrote it as \(y = (x+1)^2\).
• Switching creates \(x = (y+1)^2\).
• Solving for \(y\) gives the inverse \(y = \sqrt{x} - 1\), representing \(g(x)\).

This inverse is valid under certain conditions, based on the original function's domain and requirements for real numbers.

Function Domain

The domain of a function defines all possible input (\(x\)) values that will produce a valid output. It is often restricted by the need for real, non-negative results, particularly with square roots and other non-linear operations.
In our exercise:

• The original function \(f(x) = (x+1)^2\) has a domain of \(x \geq -1\).
• Its graph reflects over \(y = x\), creating the inverse.
• The inverse function \(g(x) = \sqrt{x} - 1\) requires a new domain of \(x \geq 0\). This ensures the square root remains real and non-negative.

Always check the inverse function's domain after transformation to ensure it aligns with regional mathematical constraints.