CBSE Class 11 Mathematics Graph of Functions Assignment Set 01

Assignment No. 6 - Relations and Functions

D - Graphs of Functions

Draw the graphs of the following functions:

Question 1. Identity Function
\(f: R \to R : f(x) = x, x \in R\)
Answer: To draw the graph of the identity function \(f(x) = x\):
1. Create a table of values by choosing a few points for \(x\) and finding corresponding values of \(f(x)\):
- When \(x = -2\), \(f(x) = -2\)
- When \(x = -1\), \(f(x) = -1\)
- When \(x = 0\), \(f(x) = 0\)
- When \(x = 1\), \(f(x) = 1\)
- When \(x = 2\), \(f(x) = 2\)
2. Plot the points \((-2, -2)\), \((-1, -1)\), \((0, 0)\), \((1, 1)\), and \((2, 2)\) on a Cartesian plane.
3. Join these points with a straight line passing through the origin at an angle of \(45^\circ\) with the positive x-axis.
In simple words: The graph of an identity function is a straight diagonal line passing through the center where every input is equal to its output.

Exam Tip: Always label the axes \(x\) and \(y\), indicate the origin \(0\), and show arrows at both ends of the line to indicate that it extends infinitely.

Question 2. Constant Function.
\(f: R \to R : f(x) = 5\) (Any Constant), \(x \in R\)
Answer: To draw the graph of the constant function \(f(x) = 5\):
1. Recognize that for any real number \(x\), the value of \(y\) (or \(f(x)\)) is always fixed at \(5\).
2. Choose several values for \(x\), for instance, \(x = -2, -1, 0, 1, 2\), which all give \(y = 5\).
3. Plot the points \((-2, 5)\), \((-1, 5)\), \((0, 5)\), \((1, 5)\), and \((2, 5)\).
4. Draw a horizontal straight line parallel to the x-axis at a height of 5 units above it.
In simple words: A constant function graph is a flat horizontal line because the output never changes no matter what input you use.

Exam Tip: Ensure the line is strictly parallel to the x-axis and intersects the y-axis exactly at the given constant value.

Question 3. Polynomial Function
\(f: R \to R : f(x) = x^2, x \in R\)
Answer: To draw the graph of the polynomial function \(f(x) = x^2\):
1. Create a table of values for \(x\) and \(f(x)\):
- When \(x = -2\), \(f(x) = 4\)
- When \(x = -1\), \(f(x) = 1\)
- When \(x = 0\), \(f(x) = 0\)
- When \(x = 1\), \(f(x) = 1\)
- When \(x = 2\), \(f(x) = 4\)
2. Plot the symmetrical points \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), and \((2, 4)\) on the coordinate plane.
3. Connect the points with a smooth, U-shaped curve pointing upwards, symmetrical about the y-axis.
In simple words: The graph of a squared function makes a symmetric bowl shape called a parabola.

Exam Tip: Do not draw a 'V' shape with sharp corners at the origin; make sure the vertex at the origin is smoothly curved.

Question 4. Rational Function
\(f: R \to R : f(x) = \frac{1}{x}, x \in R\)
Answer: To draw the graph of the rational function \(f(x) = \frac{1}{x}\):
1. Note that the function is undefined at \(x = 0\) (division by zero is not allowed).
2. Calculate values for positive and negative inputs:
- For positive x: When \(x = 0.5\), \(f(x) = 2\); when \(x = 1\), \(f(x) = 1\); when \(x = 2\), \(f(x) = 0.5\)
- For negative x: When \(x = -0.5\), \(f(x) = -2\); when \(x = -1\), \(f(x) = -1\); when \(x = -2\), \(f(x) = -0.5\)
3. Plot these points in both the first and third quadrants.
4. Sketch two separate smooth curves that approach the axes as asymptotes without touching or crossing them at \(x = 0\) and \(y = 0\).
In simple words: The reciprocal function creates two symmetric curves in opposite corners that get infinitely close to the axes but never touch them.

Exam Tip: Clearly show that the curve breaks at the y-axis and never touches either axis.

Question 5. Modulus Function
\(f: R \to R : f(x) = |x|, x \in R\)
Answer: To draw the graph of the modulus function \(f(x) = |x|\):
1. Recall the definition of absolute value: \(f(x) = x\) if \(x \ge 0\) and \(f(x) = -x\) if \(x < 0\).
2. Form a table of values:
- When \(x = -2\), \(f(x) = 2\)
- When \(x = -1\), \(f(x) = 1\)
- When \(x = 0\), \(f(x) = 0\)
- When \(x = 1\), \(f(x) = 1\)
- When \(x = 2\), \(f(x) = 2\)
3. Plot the points \((-2, 2)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), and \((2, 2)\).
4. Join the points to form a sharp V-shaped graph with its vertex at the origin \((0, 0)\).
In simple words: The absolute value graph forms a 'V' shape because all negative inputs are converted into positive outputs.

Exam Tip: The vertex at the origin must be a sharp corner (unlike the smooth curve of the parabola).

Question 6. Signum Function
\(f: R \to R : f(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}, x \in R\)
Answer: To draw the graph of the signum function:
1. Analyze the three distinct parts of the function:
- For all positive numbers (\(x > 0\)), the output is always \(1\). Draw a horizontal ray starting from just right of the y-axis at height \(1\), with an open circle at \((0, 1)\).
- At exactly \(x = 0\), the output is \(0\). Plot a solid dot at the origin \((0, 0)\).
- For all negative numbers (\(x < 0\)), the output is always \(-1\). Draw a horizontal ray starting from just left of the y-axis at height \(-1\), with an open circle at \((0, -1)\).
In simple words: The signum function graph consists of two horizontal steps at \(y = 1\) and \(y = -1\) and a single dot at the origin.

Exam Tip: Use open circles to indicate that the endpoints at the y-axis are not included for the horizontal rays, and a solid dot for the point at the origin.

Question 7. Greatest Integer Function
\(f: R \to R : f(x) = [x], x \in R\)
Answer: To draw the graph of the greatest integer function (floor function) \(f(x) = [x]\):
1. Understand that \([x]\) represents the greatest integer less than or equal to \(x\).
2. Evaluate intervals:
- For \(0 \le x < 1\), \(f(x) = 0\)
- For \(1 \le x < 2\), \(f(x) = 1\)
- For \(-1 \le x < 0\), \(f(x) = -1\)
3. Draw a series of horizontal line segments (steps) of length 1 unit.
4. Place a solid circle at the left endpoint of each step to show inclusion, and an open circle at the right endpoint to show exclusion.
In simple words: The greatest integer function graph looks like a staircase made of steps going upwards from left to right.

Exam Tip: Always alternate between solid dots on the left and hollow circles on the right for each horizontal step segment.