Problem 8 - A Textbook of Differential Calculus

Question

The range of the function \(y=\sqrt{2\\{x\\}-\\{x\\}^{2}-\frac{3}{4}}\) is (where \\{\\} denotes the fractional part) (a) \(\left[-\frac{1}{4}, \frac{1}{4}\right]\) (b) \(\left[0, \frac{1}{2}\right]\) (c) \(\left[0, \frac{1}{4}\right]\) (d) \(\left[\frac{1}{4}, \frac{1}{2}\right]\)

Step-by-Step Solution

Answer

The range of the function is \([0, \frac{1}{2}]\).

Step 1: Understanding the Fractional Part

The fractional part \( \{x\} \) represents the remainder of \( x \) after subtracting the largest integer less than or equal to \( x \). Thus, \( 0 \leq \{x\} < 1 \).

Step 2: Substituting in the Expression

Substitute \( \{x\} \) into the function to get the expression: \( y = \sqrt{2\{x\} - \{x\}^{2} - \frac{3}{4}} \).

Step 3: Finding the Valid Range for Expression

For the square root to be defined, the expression inside the root must be non-negative: \[ 2\{x\} - \{x\}^{2} - \frac{3}{4} \geq 0.\]

Step 4: Quadratic Inequality Formulation

Rearrange the inequality as a quadratic equation: \[ \{x\}^{2} - 2\{x\} + \frac{3}{4} \leq 0. \]

Step 5: Determine Roots of Quadratic

The roots of the related quadratic equation \( \{x\}^{2} - 2\{x\} + \frac{3}{4} = 0 \) can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a=1, b=-2, c=\frac{3}{4} \).

Step 6: Calculate Roots

Plug the values into the formula:\[ x = \frac{2 \pm \sqrt{4 - 3}}{2} = \frac{2 \pm 1}{2} \rightarrow x = \frac{3}{2}, \frac{1}{2}. \] Since \( \{x\} < 1 \), the range of \( \{x\} \) satisfying the inequality falls between these roots.

Step 7: Inequality Solution

The valid interval for \{x\}, which is the solution of \( \{x\}^{2} - 2\{x\} + \frac{3}{4} \leq 0 \), is \( 0.5 \leq \{x\} \leq 1 \). However, since \( \{x\} < 1 \), the effective interval is \( 0.5 \leq \{x\} < 1 \).

Step 8: Determine Range of y

Substitute the boundary conditions 1. At \( \{x\} = 0.5 \): \( y = \sqrt{2\left(0.5\right) - \left(0.5\right)^{2} - \frac{3}{4}} = 0 \) 2. At \( \{x\} = 1 \): \( y = \sqrt{2(1) - (1)^{2} - \frac{3}{4}} = 0.5 \). Thus, the range of the function is \( \left[0, 0.5\right) \).

Step 9: Finalise the Answer

Given the calculations, the range of \( y \) is closest to option (b) \( \left[0, \frac{1}{2}\right] \).

Key Concepts

Quadratic Inequality

Quadratic inequalities can seem intimidating at first, but they're a crucial concept in algebra that help us determine the range or set of values a function can take. In our exercise, the expression inside the square root must be non-negative, which leads us to the inequality \(2\{x\} - \{x\}^2 - \frac{3}{4} \geq 0\). This is essentially a quadratic inequality in terms of the fractional part \( \{x\} \), which ranges between 0 and 1.To solve the quadratic inequality, we first convert it into a standard quadratic equation: \[\{x\}^2 - 2\{x\} + \frac{3}{4} \leq 0.\]Here, we are interested in the region where this inequality holds true. By finding the roots of the related quadratic equation using the quadratic formula, we discover useful boundary points for the inequality:

• The roots are calculated from \[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot \frac{3}{4}}}{2 \cdot 1} = \frac{2 \pm 1}{2}. \]
• This yields roots at \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\).

Since \(\{x\} < 1\), we focus on \(\{x\} = 0.5\) to \(\{x\} < 1\). Understanding how quadratic inequalities limit values can lead us to significant insights about the behavior of functions.

Range of a Function

Understanding the range of a function is essential, as it tells you all possible output values. In simpler terms, the range indicates what values \(y\) can achieve given the function \(f(x)\). For our specific function \(y = \sqrt{2\{x\} - \{x\}^2 - \frac{3}{4}}\), we're interested in the interval of \(y\) values when the expression inside the square root evaluates to non-negative.To find the range, we first need to figure out the valid values of \(\{x\}\). With \(\{x\}\) calculated from the inequality steps:- We determine \(0.5 \leq \{x\} < 1\).We analyze the boundary conditions at these extremes:

• When \(\{x\} = 0.5\), the expression inside the square root becomes 0, so \(y = \sqrt{0} = 0\).
• As \(\{x\} \) approaches 1, we evaluate \(y = \sqrt{2(1) - (1)^2 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = 0.5\).

Thus, the range of \(y\) becomes \([0, 0.5)\). Knowing how to determine these intervals is a fundamental skill for handling functions efficiently.

Square Root Function

Square root functions are pivotal in algebra, displaying a special behavior compared to other polynomial functions. The square root function generally takes the form of \(y = \sqrt{g(x)}\), where we must ensure that the expression \(g(x)\) is non-negative so that \(y\) remains a real number.In our problem, the function is \(y = \sqrt{2\{x\} - \{x\}^2 - \frac{3}{4}}\). Some key points about square root functions are:

• They are defined only for \(g(x) \geq 0\).
• If \(g(x) \) becomes negative, \(y\) becomes undefined in the set of real numbers.

The square root function's domain is determined by the values of \(x\) that satisfy \(g(x) \geq 0\). Consequently, understanding both the behavior and constraints imposed by the square root opens up pathways to solve a variety of mathematical problems, giving us clarity on how the function behaves for different inputs.