CBSE Class 11 Mathematics Graph of Functions Assignment Set A answers

Sketching of the graphs

1. Identity Function: \(f(x) = x, x \in R\)
Answer: Let \(y = x\).
Table of values:
\(x = 1, 2, 3\)
\(y = 1, 2, 3\)
The graph is a straight line passing through the origin \((0, 0)\) making an angle of \(45^\circ\) with the positive direction of the x-axis.
In simple words: The identity function outputs the exact same value that is inputted, forming a diagonal line straight through the graph.

Exam Tip: Remember that the domain and range of the identity function are both equal to the set of all real numbers \(R\).

2. Constant Function: \(f(x) = \text{Constant} = 5 \text{ (let)}\) (or any number), \(x \in R\)
Answer: Let \(y = 5\).
For any real value of \(x\), the value of \(y\) remains fixed at \(5\).
The graph is a horizontal line parallel to the x-axis at a distance of \(5\) units above it.
In simple words: No matter what number you put in for \(x\), the output \(y\) is always the same constant value.

Exam Tip: The graph of a constant function is always a horizontal line, and its range contains only a single element.

3. Polynomial Function: \(f(x) = x^2, x \in R\)
Answer: Let \(y = x^2\).
Table of values:
\(x = 0, 1, -1, 2, -2, 3, -3, 4, -4\)
\(y = 0, 1, 1, 4, 4, 9, 9, 16, 16\)
The graph is a parabola symmetric about the y-axis with vertex at the origin \((0, 0)\).
In simple words: Squaring negative and positive numbers gives positive outputs, creating a U-shaped curve called a parabola.

Exam Tip: The domain is \(R\) and the range is \([0, \infty)\) because the square of any real number is never negative.

4. Rational Function: \(f(x) = \frac{1}{x}, x \in R\)
Answer: Let \(y = \frac{1}{x}\).
Table of values:
\(x = 1, -1, 2, -2, 1.5, -0.5, 0.25, 0.5, 1.5\)
\(y = 1, -1, 0.5, -0.5, -0.67, -2, 4, 2, 0.67\)
Note that \(x \neq 0\) since division by zero is undefined.
The graph consists of two branches in the first and third quadrants, forming a rectangular hyperbola.
In simple words: As \(x\) gets closer to zero, \(y\) shoots off towards infinity, and as \(x\) gets very large, \(y\) approaches zero.

Exam Tip: Always state that \(x \neq 0\) when defining the domain of this rational function.

5. Modulus Function: \(f(x) = |x|, x \in R\)
Answer: Let \(y = |x|\).
By definition:
\(f(x) = x\) if \(x \geq 0\)
\(f(x) = -x\) if \(x < 0\)
For \(x \geq 0\), \(y = x\) (a line in the first quadrant).
For \(x < 0\), \(y = -x\) (a line in the second quadrant).
The graph is V-shaped with its vertex at the origin \((0, 0)\).
In simple words: The modulus function strips away any negative sign, turning all inputs into their positive values.

Exam Tip: The graph looks like the letter V, and the range is always the set of non-negative real numbers.

6. Signum Function: \(f(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}, x \in R\)
Answer: The signum function extracts the sign of a real number.
When \(x > 0\), \(y = 1\).
When \(x = 0\), \(y = 0\).
When \(x < 0\), \(y = -1\).
The graph consists of three parts: a ray at \(y = 1\) for positive x, a single point at the origin \((0, 0)\), and a ray at \(y = -1\) for negative x.
In simple words: It tells you whether a number is positive (gives 1), zero (gives 0), or negative (gives -1).

Exam Tip: Pay close attention to open circles at the endpoints of the horizontal rays when sketching the signum function.

7. Greatest Integer Function: \(f(x) = [x], x \in R\)
Answer: Let \(y = [x]\). By definition, the greatest integer function gives the greatest integer less than or equal to \(x\).
If \(y = [x] = -1\), then \(-1 \leq x < 0\)
If \(y = [x] = 0\), then \(0 \leq x < 1\)
If \(y = [x] = 1\), then \(1 \leq x < 2\)
If \(y = [x] = 2\), then \(2 \leq x < 3\)
If \(y = [x] = 3\), then \(3 \leq x < 4\), and so on.
The graph consists of a series of step-like horizontal line segments.
In simple words: It rounds down any number to the nearest integer below or equal to it, creating a staircase graph.

Exam Tip: Use solid dots for left endpoints and hollow circles for right endpoints on each step of the staircase graph.