Problem 8 - A Complete Resource Book in Mathematics for JEE Main

Question

If the functions \(f, g, h\) are defined from the set of real numbers \(R\) to \(R\) such that \(f(x)=x^{2}-1, g(x)=\sqrt{x^{2}+1}, h(x)=\left\\{\begin{array}{l}0, \text { if } x \leq 0 \\ x, \text { if } x \geq 0\end{array}\right.\) then the composite function (hofog) \((x)=\) (A) \(\begin{cases}0, & x=0 \\ x^{2}, & x>0 \\ -x^{2}, & x<0\end{cases}\) (B) \(\begin{cases}0, & x=0 \\ x^{2}, & x \neq 0\end{cases}\) (C) \(\begin{cases}0, & x \leq 0 \\ x^{2}, & x>0\end{cases}\) (D) None of these

Step-by-Step Solution

Answer

Option C: \(\begin{cases}0, & x \leq 0 \\ x^2, & x > 0\end{cases}\).

Step 1: Understand Each Function

First, we examine each individual function: \(f(x) = x^2 - 1\), which is a quadratic function; \(g(x) = \sqrt{x^2 + 1}\), which outputs the non-negative square root of \(x^2 + 1\); and \(h(x)\), which is a piecewise function that equals 0 when \(x \leq 0\) and \(x\) when \(x > 0\).

Step 2: Compute the Inner Composite Function \(g(f(x))\)

Let's find \(g(f(x))\). First, calculate \(f(x) = x^2 - 1\). Substitute in \(g\): \[g(f(x)) = g(x^2 - 1) = \sqrt{(x^2 - 1)^2 + 1}\,\] resulting in \(g(f(x)) = \sqrt{x^4 - 2x^2 + 1 + 1} = \sqrt{x^4 - 2x^2 + 2}\).

Step 3: Compute the Composite Function \(h(g(f(x)))\)

Knowing \(h(x)\) is defined as a piecewise function, we need to determine the values to substitute: - If \(g(f(x)) \leq 0\), then \(h(g(f(x))) = 0\)- If \(g(f(x)) > 0\), then \(h(g(f(x))) = g(f(x))\).Since \(g(f(x)) = \sqrt{x^4 - 2x^2 + 2}\) is always non-negative and positive, we analyze it for different \(x\) values.

Step 4: Analyze \(h(g(f(x)))\)

Investigate the outputs based on \(x\):- For \(x \leq 0\): Since zero leads to non-negative outputs, \(h(g(f(x))) = 0\) due to the condition \(h(\text{non-positive}) = 0\).- For \(x > 0\): \(g(f(x))\) is positive, thus \(h(g(f(x))) = g(f(x)) = \sqrt{x^4 - 2x^2 + 2}\). Consequently, \(h(g(f(x))) = x^2\) for positive \(x\) since \(x^2\) is the square of \(x\).

Step 5: Match with Options

From Step 4, determine the outcome: \[\begin{cases}0, & x \leq 0 \ x^2, & x > 0\end{cases}\,\] which matches option C.

Key Concepts

Quadratic Function

Quadratic functions are a staple in mathematics. These functions are represented in the standard form: \( f(x) = ax^2 + bx + c \). The function given in this problem, \( f(x) = x^2 - 1 \), is a simple quadratic function where \( a = 1 \), \( b = 0 \), and \( c = -1 \). Quadratic functions are known for their parabolic graphs, which can open upwards or downwards depending on the sign of \( a \).

Key features of quadratic functions include:

• The vertex: For \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \).
  
• The axis of symmetry: This is a vertical line through the vertex, also represented by \( x = -\frac{b}{2a} \).
• The intercepts: The y-intercept is found by evaluating \( f(0) \), and x-intercepts can be found by solving \( f(x) = 0 \).

In the context of composite functions, the quadratic function often serves as the initial transformation of input values, setting the stage for further transformations through other types of functions.

Piecewise Function

Piecewise functions are interesting because they are defined by different expressions based on different intervals of the input variable. In this problem, the function \( h(x) \) is such a function, defined piecewise as:
\[h(x) = \begin{cases}0, & \text{if } x \leq 0 \x, & \text{if } x > 0\end{cases}\]
Piecewise functions can represent real-world situations where different rules apply in different scenarios. For example, h(x) could represent a physical process that behaves differently depending on conditions, such as temperature thresholds or time intervals.

When analyzing piecewise functions:

• Determine the range of each piece by solving inequalities for transition points.
• Evaluate limits at points where the definition changes to comprehend continuity or discontinuity.
• Apply each piece formula appropriately depending on the input value.

In composite functions, piecewise functions can be effectively used to capture distinct outputs based on subsets of input, making them versatile and powerful in mathematical modeling.

Square Root Function

Square root functions involve the expression \( g(x) = \sqrt{x} \), or in the problem's case, \( g(x) = \sqrt{x^2 + 1} \). The square root function is always defined for non-negative inputs, producing non-negative outputs, and it is essential for determining particular solutions when evaluating composite functions.

Key properties of square root functions include:

• Domain: Generally \( x \geq 0 \) for \( \sqrt{x} \), but more can be if under a square like in \( \sqrt{x^2 + 1} \).
• Shape: A smooth curve starting at the origin when \( x = 0 \), gradually increasing.
• Asymptotic behavior: Approaches horizontal as \( x \to \infty \).

Composite functions involving square roots must ensure the radicand (the expression under the square root) is non-negative. This aspect makes them quite significant in theoretical and applied mathematics as it introduces domain restrictions and implications for continuity and differentiability. In the problem at hand, \( g(f(x)) \) requires particular attention to these details to ensure valid outputs through the composite function.