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Modulus function gives us the magnitude of the number i.e the absolute value of the number. Another name for the modulus function is the Absolute Value Function. The basic rule for this function in the simplest language is that it always returns a positive answer whatever variable is given as the input to the function. The modulus function in arithmetic is denoted by |x| and is also referred to as the distance between the number and 0 or its origin.
Also Read: Onto Function
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Keyterms: Absolute value, Modulus Function, Domain, range, Number Line, Straight Line, Real Number, Function, Variable
Also Read: Complex Numbers and Quadratic Equations
Definition of the Modulus Function
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The modulus function is defined as the function of a variable ‘a’ where f(a) = |a|.
- If the value of a is positive then the value of f(a) is ‘a’ itself.
- If the value of a is negative, then the value of f(a) is ‘- (a)’.
- Therefore, the formula definition for this function should be:
- f(a) = { a where a >= 0 }
- f(a) = -a where a < 0 } where a ∈ R
Also Read: Relations and Functions Ncert Solutions
Graph of the Modulus Function
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Let us understand the logic behind the graph of the modulus function before we see how to draw it. Let us discuss about the domain and range of the function:
- Domain - The values you input into the function. The value can be any real number. Therefore, the domain is R.
- Range - The values you get as output from the function. The values are always positive. Therefore, the range is [0,∞).
- Therefore, the graph should be a straight line inclined at 45° on the positive side of the x-axis and a straight line inclined at -45° on the negative side of the x-axis.
Also Read: Relations and Functions Important Questions
Things to Remember
- The graph is fairly simple to remember and an easy way to remember is thinking of it as the letter ‘V’ with the base of it in the center of the graph at (0,0).
- The modulus function gives the value of the number/variable by making its sign positive if negative and the number itself if positive. For 0, it is simply 0.
- Modulus is also referred to as the distance from 0 on the number line. Distance is always positive, displacement can be negative.
- The domain of a modulus function is any real number while the range of a modulus function is all numbers from 0 to infinity including 0. That’s why it is represented with a square bracket → [0,∞)
- When solving for the value of a variable in a question, after getting the values, always put the values in the equations and solve the equations to verify that no calculation mistakes have been committed.
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Sample Questions
Ques 1. Find the value of the modulus function |x| for x = -5 and x = 10 (2 marks)
Ans. The formula for modulus function is |x| = x when x >= 0 and |x |= - x when x < 0
Therefore, |x| = -(-5) = 5 when x = -5 and |x| = 10 when x = 10
Ques 2. Solve | x + 12 | = 9 and verify your solution. (4 marks)
Ans. The formula for modulus function is |x| = x when x >= 0 and |x |= - x when x < 0
So, we will solve the equation with both values :
Equation 1: x + 12 = 9
x = 9-12
x = -3
Equation 2: x + 12 = -9
x = -9 - 12
x = -21
We can also verify our answer by putting the values in the equation like:
For the value of x = -3,
| -3 + 12 | = 9
| 9 | = 9
9=9
L.H.S =R.H.S
Hence Proved.
For the value of x = -21,
| -21 + 12 | = 9
| -9 | = 9
9=9
L.H.S =R.H.S
Hence Proved.
Ques 3. For the function y = | x - 3 |. What is y(-1)? (2 marks)
Ans. The formula for modulus function is |x| = x when x >= 0 and |x | = - x when x < 0
So, we will solve the equation with x = -1 :
y = | x - 3 |
y = | -1 - 3 |
y = | -4 |
y = 4.
Therefore, y(-1) = 4.
Q.E.D
Ques 4. Solve | x | < 6 |, | 6x | > 24, | x | <= - 7 and | x - 15 | + 6 < 7 (4 marks)
Ans. The formula for modulus function is |x| = x when x >= 0 and |x |= - x when x < 0
| x | < 6 → x < 6 and x > -6
-6 < x < 6
Therefore, x ∈ (-6,6)
| 6x | > 24
| x | > 4 → x > 4 and x < - 4
Therefore, ( - ∞, - 4 ) ? [ 4, ∞ )
| x | <= - 7
The modulus of any function can never be a negative number. Therefore, this has no solution.
| x - 15 | + 6 < 7
| x - 15 | < 1 → x - 15 < 1 and x - 15 > - 1
- 1 < x - 15 < 1
Adding 15 on all sides
- 1 + 15 < x - 15 + 15 < 1 + 15
14 < x 16
Therefore, x ∈ (14, 16)
Ques 5. If f(x) = 3 | x |+ 2 is modulus, then plot the graph for this function. (4 marks)
Ans. The formula for modulus function is |x| = x when x >= 0 and |x |= - x when x < 0
So, for x < 0 → f(x) = -3x + 2 and for x >=0 → f(x) = 3x + 2
x < 0
| x | -1 | -2 |
| f(x) | 5 | 8 |
| x | 0 | 1 | 2 |
| f(x) | 2 | 5 | 8 |
Ques 6. Solve | x + 6 | = 18 and verify your solution. (4 marks)
Ans. The formula for modulus function is |x| = x when x >= 0 and |x |= - x when x < 0
So, we will solve the equation with both values :
Equation 1: x + 6 = 18
x = 18-6
x = 12
Equation 2: x + 6 = -18
x = -18 - 6
x = -24
We can also verify our answer by putting the values in the equation like:
For the value of x = 12,
| 12 + 6 | = 18
| 18 | = 9
18=9
L.H.S =R.H.S
Hence Proved.
For the value of x = -24,
| -24 + 6 | = 18
| -18 | = 18
18=18
L.H.S =R.H.S
Hence Proved.
Ques 7. Find domain and range of y = | 3 - x | (2 marks)
Ans. Clearly, the domain can be any real number.
Therefore, the domain is R.
For range, | 3-x | is always greater than or equal to 0.
Thus, the range is [0, ∞).
Ques 8. For y = | x + 2 |. Find y(-3) and y(4) (3 marks)
Ans. The formula for modulus function is |x| = x when x >= 0 and |x | = - x when x < 0
So, we will solve the equation with x = -3 :
y = | x + 2 |
y = | - 3 + 2 |
y = | -1 |
y = 1.
Therefore, y(-3) = 1.
Next, we will solve the equation with x = 4 :
y = | x + 2 |
y = | 4 + 2 |
y = | 6 |
y = 6.
Therefore, y(4) = 6
Q.E.D
Ques 9. Find the domain for y =x | x | and draw its graph. (3 marks)
Ans. Clearly, the domain can be any real number.
Therefore, the domain is R.
The graph can be plotted as:
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