Problem 149 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: 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 \(g(x)=\) \(\sqrt{x}-1, x \geq 0\) Reason: \(g(x)\) is the inverse of \(f(x)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The problem is asking us to determine the function \(g(x)\), which is the reflection of \(f(x)\) about the line \(y = x\). This is equivalent to finding the inverse of \(f(x)\).

Step 2: Determine if \(f(x)\) is One-to-One

Check if \(f(x) = (x+1)^2\) for \(x \geq -1\) is a one-to-one function to ensure it has an inverse. Since \(f(x)\) is a parabola opening upwards starting from \((-1,0)\), and \(f(x)\) is strictly increasing on its domain, \(f(x)\) is one-to-one.

Step 3: Find the Inverse Function of \(f(x)\)

To find the inverse, let's solve \(y = (x+1)^2\) for \(x\):- Rewrite the equation: \(y = (x+1)^2\).- Solve for \(x\): \(x + 1 = \sqrt{y}\) (since \(x+1\) is non-negative on its domain, we take the positive root).- \(x = \sqrt{y} - 1\).Therefore, the inverse function is \(g(x) = \sqrt{x} - 1\) with \(x \geq 0\) to ensure the domain of \(g(x)\) matches the range of \(f(x)\).

Step 4: Verify the Function and its Inverse

Verify that \(f(g(x)) = x\) and \(g(f(x)) = x\) to confirm that the inverse is correct.- For \(f(g(x))\): \(f(g(x)) = f(\sqrt{x} - 1) = (\sqrt{x} - 1 + 1)^2 = (\sqrt{x})^2 = x\).- For \(g(f(x))\): \(g((x+1)^2) = \sqrt{(x+1)^2} - 1 = x+1 - 1 = x\). Both verifications confirm that \(g(x)\) is indeed the inverse of \(f(x)\).

Step 5: Conclusion

The function \(g(x) = \sqrt{x} - 1\) is the reflection of \(f(x)\) across the line \(y = x\), confirming the assertion that \(g(x)\) is the inverse of \(f(x)\).

Key Concepts

Function Reflection

When working with functions, an important concept is that of **reflection**. In particular, reflecting a function across the line \( y = x \). This type of reflection involves swapping the roles of \( x \) and \( y \) to create the function’s inverse. A reflection across this line can be thought of as flipping the function over the diagonal where \( x = y \). For example, if you have a graph and you draw a diagonal line at \( y = x \), the reflection would mean every point on your graph 'mirrors' this diagonal.
To better understand, consider the function \( f(x) = (x + 1)^2 \). Instead of looking at \( y = f(x) \), you reflect it to find the inverse function, by solving for the variable \( x \) in terms of \( y \).

• This yields the inverse \( g(x) = \sqrt{x} - 1 \) since solving for \( x \) gives you the function in which the roles have been swapped.
• This inverse, \( g(x) \), is the reflection of \( f(x) \) across the line \( y = x \).
• The reflection inherently moves coordinates \((x, y)\) to \((y, x)\).

Understanding reflection is key to grasping why finding inverses often involves a switch in variables.

One-to-One Function

A **one-to-one function** is crucial to ensuring that a function has an inverse. But what does one-to-one mean, exactly? Simply put, a function is one-to-one if every distinct input \( x \) maps to a distinct output \( y \). Conversely, no two different inputs result in the same output.
This concept is essential for functions with inverse functions because only one-to-one functions can have valid inverses that are also functions. Visualizing it helps: If you drew a horizontal line across the graph and it crosses at more than one point, the function is not one-to-one. In mathematical terms, we say:\[ f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \]
For the function \( f(x) = (x + 1)^2 \) where \( x \geq -1 \), it is important to notice that this is a part of a parabola, and within the domain specified, it is strictly increasing.

• This restriction to \( x \geq -1 \) ensures that there are no repeated \( y \) values for different \( x \) values, confirming that it's one-to-one.
• Because of this, \( f(x) \) qualifies to find an inverse, specifically \( g(x) = \sqrt{x} - 1 \).

Recognizing one-to-one functions is foundational to inverse operations in mathematics.

Finding Inverse Functions

Finding the inverse of a function is a typical exercise in algebra, often involving a multi-step process. Firstly, ensure the function is one-to-one, as previously discussed. Once confirmed, finding the inverse helps understand the function's reflection across \( y = x \). To find an inverse function:

• Replace \( f(x) \) with \( y \) for clarity.
• Switch the roles of \( x \) and \( y \). This means you treat \( y \) as a function of \( x \).
• Solve for \( y \), which now becomes the new function in terms of \( x \).

In our problem:
You start with \( y = (x+1)^2 \). Swapping gives us \( x = (y+1)^2 \). Solving this for \( y \), we find \( y = \sqrt{x} - 1 \).
This is \( g(x) = \sqrt{x} - 1 \), the inverse function.

• It's important to always check back; verify by confirming that applying \( f \) to \( g(x) \) and vice versa returns the original \( x \).
• Verification proves our inverse determination.
  Inverses are valuable for understanding relationships in functions, as well as practical uses, such as finding original values from transformed data.

Let's summarize: inverses reflect functions about \( y=x \), require one-to-one functions, and are found through careful algebraic manipulations.