Problem 83 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(g(x)=1+\sqrt{x}\) and \(f[g(x)]=3+2 \sqrt{x}+x\), then \(f(x)=\) \(\begin{array}{llll}\text { (A) } 1+2 x^{2} & \text { (B) } 2+x^{2} & \text { (C) } 1+x & \text { (D) } 2+x\end{array}\)

Step-by-Step Solution

Answer

f(x) = x^2 + 2, which corresponds to option B.

Step 1: Understand the Function Composition

We need to find the expression for \( f(x) \) given that \( f(g(x)) = 3 + 2\sqrt{x} + x \). The function \( g(x) = 1 + \sqrt{x} \) is given. Our goal is to express \( f(x) \) in terms of a variable \( u \) where \( u = g(x) \). This means substituting into the known \( f(g(x)) \) expression.

Step 2: Substitute \( g(x) \) into \( f[g(x)] \)

Since \( u = g(x) = 1 + \sqrt{x} \), we need \( f(1 + \sqrt{x}) = 3 + 2\sqrt{x} + x \). Identify that \( \sqrt{x} = g(x) - 1 \), hence \( x = (g(x) - 1)^2 \). Let's substitute \( \sqrt{x} \) in terms of \( u \).

Step 3: Express \( f(u) \) in terms of \( u \)

Given that \( g(x) = u = 1 + \sqrt{x} \), this implies that \( \sqrt{x} = u - 1 \) and \( x = (u-1)^2 \). Substitute \( u \) into the expression \( f(g(x)) = 3 + 2(u-1) + (u-1)^2 \).

Step 4: Simplify the Expression

Expand and simplify the expression: \( 3 + 2(u-1) + (u-1)^2 \) becomes:\[3 + 2(u - 1) + (u - 1)^2 = 3 + 2u - 2 + (u^2 - 2u + 1)\]Simplify it further: \( 1 + 2u - 1 + u^2 - 2u + 1 = u^2 + 2 \).

Step 5: Arrive at Final Function \( f(x) \)

From the previous simplification, we find that the expression reduces to \( u^2 + 2 \). Since \( u = x \) for \( f(x) \), we have \( f(x) = x^2 + 2 \).

Key Concepts

Substitution Technique

Substitution technique is a powerful tool in algebra that helps us solve complex problems by simplifying expressions. This method involves replacing a variable or an expression with another expression that makes the problem easier to solve.
In our original problem, the substitution technique is applied by introducing a new variable, \(u\), to simplify the expression for \( f(g(x)) \).
This involves holding \( u = g(x) \) and using it to represent other complex parts of the expression.

• Firstly, identify what part of the expression needs simplification, such as transforming \( g(x) = 1 + \sqrt{x} \).
• Next, express the elements involved in terms of the new variable, \( u \). For example, \( \sqrt{x} \) becomes \( u-1 \).
• Then, the substitution helps us transform an originally complicated function into an easier one: \( x = (u-1)^2 \) in this scenario.

Using substitution makes it straightforward to reformulate and solve expressions, breaking them down into manageable parts.

Composite Functions

Composite functions are simply functions formed by the combination of two functions. If you have two functions, say \( f \) and \( g \), you can create a composite function \( f(g(x)) \) by applying one function to the result of another.
Understanding composite functions is crucial when dealing with function composition problems as seen in our exercise.
When looking at \( f(g(x)) = 3 + 2\sqrt{x} + x \), it implies that \( g(x) \) is the input that flows first into \( f \). This requires understanding how each function manipulates and alters the input.

• Recognize \( g(x) \) as an outer layer that transforms \( x \) before passing it to \( f \).
• The key step is aligning \( f(x) \) correctly by ensuring that whatever \( g(x) \) leaves off as output becomes the exact input for \( f \).
• This concept enables blending and manipulating functions into a seamless operation that efficiently outputs a new value or expression.

Composite functions allow better control and versatility when structuring calculations and modeling complex systems.

Algebraic Expressions

Algebraic expressions are fundamental in mathematics, consisting of numbers, variables, and arithmetic operations arranged together.
They form the building blocks that allow us to create equations and functions, like those seen in calculus and algebra.
In the initial exercise, understanding the makeup and manipulation of algebraic expressions is crucial.

• Expressions like \( 1 + \sqrt{x} \) and \( 3 + 2\sqrt{x} + x \) are manipulated using algebraic rules to form new expressions.
• Simplification and transformation using expressions are applied heavily to reach conclusions, such as substituting \( \sqrt{x} \) with \( u-1 \).
• Throughout algebra and function composition, recognizing patterns in expressions and how they interact through operations is key to solving for \( f(x) \).

Getting comfortable with rearranging, expanding, and simplifying algebraic expressions deepens understanding and problem-solving skill, making this concept indispensable in mathematics.