Problem 119 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The distinct linear function which maps \([-1,1]\) onto \([0,2]\) is (A) \(x-1\) (B) \(x+1\) (C) \(-x+1\) (D) \(-x-1\)

Step-by-Step Solution

Answer

The correct option is (B) \(x+1\).

Step 1: Understand the Problem

We need to find a linear function that maps the interval \([-1,1]\) onto the interval \([0,2]\). This means the function should take \(-1\) to \(0\) and \(1\) to \(2\).

Step 2: General Form of Linear Function

A linear function is generally represented as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 3: Determine the Slope

We use the formula for the slope \(m\) between two points \((x_1, y_1), (x_2, y_2)\): \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{1 - (-1)} = \frac{2}{2} = 1\]

Step 4: Determine the Y-intercept

Using the point \((-1, 0)\) and slope \(m = 1\), plug these into the equation \(y = mx + b\):\[0 = 1(-1) + b \Rightarrow 0 = -1 + b \Rightarrow b = 1\]

Step 5: Formulate the Linear Function

Substitute \(m = 1\) and \(b = 1\) back into the linear function to get:\[y = x + 1\]

Step 6: Verify the Function

Check if the function \(y = x + 1\) satisfies the conditions: - For \(x = -1\), \(y = -1 + 1 = 0\)- For \(x = 1\), \(y = 1 + 1 = 2\)This confirms the correct mapping.

Key Concepts

Slope Calculation

In the context of a linear function, understanding the slope is crucial. The slope represents the rate of change or steepness of a line, often described as "rise over run." To calculate it, you'll look at two endpoint coordinates and use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

For example, when mapping the interval \([-1, 1]\) onto \([0, 2]\), the coordinates given are \((-1, 0)\) and \((1, 2)\). By plugging these numbers into the slope formula, you obtain \[ m = \frac{2 - 0}{1 - (-1)} = \frac{2}{2} = 1 \].

This calculation indicates that for every unit increase in \(x\), \(y\) also increases by one unit, confirming a direct linear relationship. A positive slope means the function increases, while a negative slope shows a decrease. This basic understanding of slope encourages you to visualize how steep or flat a particular line may be.

Linear Equation

Understanding a linear equation is fundamental to solving problems involving straight lines. A linear equation is usually expressed in the form \( y = mx + b \). In this equation, \(m\) is the slope, and \(b\) is the y-intercept, which indicates where the line crosses the y-axis.

In the step-by-step solution provided, once we determined the slope \(m = 1\), the next step was identifying the y-intercept \(b\). By using a known point \((-1, 0)\), the equation becomes:

• \(0 = 1(-1) + b \)
• \(0 = -1 + b \)
• \(b = 1 \)

Thus, the linear equation becomes \( y = x + 1 \).

This equation tells us that the line begins at \(y = 1\) on the y-axis, and increases in a direct, linear fashion with unit increments in \(x\). Understanding how to formulate a linear equation is key to connecting real-world data points or solving theoretical math problems.

Interval Mapping

Interval mapping involves translating one set of values smoothly into another set, ensuring each input has a proper corresponding output. For linear functions, such mapping signifies that the function should consistently transform values between specified intervals.

In this exercise, we want to map \([-1, 1]\) onto \([0, 2]\). The chosen linear function should ensure that every point in \([-1, 1]\) — which denotes inputs of the function — corresponds directly to a point in \([0, 2]\). By verifying:

• \(x = -1 \rightarrow y = 0 \)
• \(x = 1 \rightarrow y = 2 \)

For the function to be correct, these endpoints must be mapped accurately, as exhibited by \( y = x + 1 \).

Interval mapping is a useful concept in contexts where boundaries and transformations of values are significant, like converting measurements or adjusting scales. Knowing correct mappings facilitates ensuring accuracy in these transformations.