Problem 11 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Construct the graph of the function given below $$ \begin{aligned} f(x) &=x-1, & x<0 \\ &=\frac{1}{4}, & x &=0 \\ &=x^{2}, & x>0 . \end{aligned} $$

Step-by-Step Solution

Answer

To construct the graph of the given piecewise function, follow these steps: 1. Plot the line \(f(x) = x-1\) for \(x < 0\) with an open circle at (0, -1). 2. Include a closed circle at the point (0, 1/4) for \(x = 0\). 3. Plot the right half of the parabola \(f(x) = x^2\) for \(x > 0\) starting at the point (0, 0) and opening upwards. Combine the three pieces to form the complete graph of the function.

Step 1: Plot the graph for \(x < 0\)

Plot the line \(f(x) = x-1\) only for the values in the range \(x < 0\). This is a straight line with a slope of 1 and a y-intercept of -1.

Step 2: Plot the graph for \(x = 0\)

Since \(x = 0\) is a single point, we only have one point to plot on the graph. For \(x = 0\), the function is defined as \(f(0) = \frac{1}{4}\). Plot the point (0, 1/4).

Step 3: Plot the graph for \(x > 0\)

For \(x > 0\), the function is defined as \(f(x) = x^2\). This is a parabola with a vertex at (0, 0) and opens upwards. However, because our function is only defined for \(x > 0\), we will only plot the right half of the parabola.

Step 4: Combine the Graphs

To complete the graph of the given piecewise function, we will now combine the three graphs from Steps 1, 2, and 3: 1. The line for \(x < 0\) should extend to the left but not include the point at \(x = 0\) (open circle on the point (0, -1)). 2. The point at \(x = 0\) should be included on the graph as a closed circle (because the function is defined at this point). 3. Finally, the parabola for \(x > 0\) should start exactly at the point (0, 0) and open upwards. Now you would have a complete graph of the given piecewise function.

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in mathematics that allows you to visually represent how these functions behave.
When graphing piecewise functions like the one in our exercise, you must consider each segment separately and then combine them into a single graph.
This particular function consists of three separate parts:

• A linear function: For this part, graphing involves plotting a straight line. You determine the slope and the y-intercept and then plot points accordingly. For example, the line defined by \( f(x) = x - 1 \) for \( x < 0 \) is graphed as a line extending to the left from its y-intercept of -1.
• A constant function: This is represented by a single point, such as \( f(x) = \frac{1}{4} \) at \( x = 0 \). Here, you'll just plot a single dot on the graph, making sure it is closed to indicate the value exists at this point.
• A quadratic function: For \( x > 0 \), this parabolic curve is illustrated by plotting points that fit \( f(x) = x^2 \). You'll capture half of this parabola extending upwards from the origin (0,0).

Once all parts are graphed, ensure that they connect smoothly according to the conditions given in the piecewise function.

Functions and Graphs

Understanding functions and their graphs is crucial in math, as they show how variables interact and change.
Piecewise functions add complexity because they use different rules over different interval domains.

• Every part of the graph is defined under its own conditions based on the domain specified. For instance, the linear segment and the parabolic segment each cover different sections of the x-axis.
• Each segment may have different continuity. Closed circles in the graph mean the function is defined there, as with the point at \( x = 0 \). Open circles are used when a value is not included, demonstrating a break or gap in continuity.
• The transition between these segments often highlights functions’ behavior changes, significant for understanding local behaviors and limits.

Graphs visually demonstrate these variances using distinct sections to help outline function behavior clearly and define these areas accurately.

Differential Calculus

Differential calculus is a branch of mathematics focusing on how functions change continuously. It primarily deals with concepts like derivatives that measure rates of change.
While the actual derivative calculation isn't directly demonstrated in graphing a piecewise function, the understanding of calculus principles aids in analyzing the changes in slopes.

• For lines, like \( f(x) = x - 1 \), the slope or derivative is constant. This steady inclination indicates a consistent rate of change.
• The function \( f(x) = x^2 \) is non-linear with increasing change, indicative of a variable rate—captured by its changing slope in the graph.
• By applying differential calculus, you explore how each piece of the function behaves individually and collectively.

Whether analyzing just one segment or the complete function, concepts from differential calculus can provide deeper insight into understanding how different types of functions shift, providing a broader comprehension of mathematical behavior over specified intervals.