Question 28 of 33intermediate🔧 ApplyShort Answer2 marks

Find the domain and the range of the real function ff defined by f(x)=∣x−1∣f(x) = |x - 1|.

Correct Answer

Domain of f: Since the modulus function is defined for each real number, the function f(x)=∣x−1∣f(x) = |x - 1| is defined for all real numbers. Therefore, Domain = R (all real numbers).

Range of f: The modulus function always gives non-negative values. For any x∈Rx \in R, we have ∣x−1∣≥0|x - 1| \geq 0. The minimum value of ∣x−1∣|x - 1| is 0 (attained when x=1x = 1), and it can take any positive real value. Therefore, Range = {y : y ≥\geq 0} or [0,∞)[0, \infty).

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 5 | (Page 17)
For More Understanding

Explanation

The question relates to the modulus function described in the textbook context. The context states that the modulus function f(x)=∣x∣f(x) = |x| is defined for each x∈Rx \in R (domain is all real numbers) and gives non-negative values. For f(x)=∣x−1∣f(x) = |x - 1|, this is simply a horizontal shift of the basic modulus function. The domain remains all real numbers since the expression inside the modulus is defined for all xx. The range remains non-negative since the modulus operation always yields values ≥0\geq 0.

Solution Steps

  1. Step 1: Identify that f(x)=∣x−1∣f(x) = |x - 1| is a modulus function (shifted form of ∣x∣|x|).

  2. Step 2: Since modulus function is defined for all real numbers, Domain = R.

  3. Step 3: Since modulus function gives non-negative values, ∣x−1∣≥0|x - 1| \geq 0 for all xx.

  4. Step 4: Therefore, Range = {y : y ≥\geq 0} = [0,∞)[0, \infty).

PreviousNext