A real valued function f is defined as f(x)=-1 for -2≤ x≤ 0 and f(x)=x-1 for 0<x≤ 2. Which of the following statements is FALSE?

C. f(|x|)+|f(x)|=1, if 0≤ x≤ 1. Step 1: For 0≤ x≤ 1: f(|x|)=f(x)=x-1≤ 0 and |f(x)|=1-x. Step 2: So f(|x|)+|f(x)|=(x-1)+(1-x)=0≠ 1. Statement (C) is false. Step 3: The others hold: on [1,2], f(x)=x-1≥ 0, so |f(x)|=x-1 and f(|x|)-|f(x)|=0; on [0,1], f(|x|)=|x|-1. Check: x=1/2: f(1/2)=-1/2, so f(|x|)+|f(x)|=-1/2+1/2=0, not 1. Trap: (D) looks false if you expect the modulus to flip the sign, but on [1,2], x-1≥ 0, so |f(x)|=f(x) and the difference is 0.

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A real valued function f is defined as $f (x) = - 1$ for $- 2 \leq x \leq 0$ and $f (x) = x - 1$ for $0 < x \leq 2$ . Which of the following statements is FALSE?

Answer (Detailed Solution Below)View answer ▾

Answer: Option C — $f (∣ x ∣) + ∣ f (x) ∣ = 1$ , if $0 \leq x \leq 1$

Detailed Solution

For $0 \leq x \leq 1$ : $f (∣ x ∣) = f (x) = x - 1 \leq 0$ and $∣ f (x) ∣ = 1 - x$ .

So $f (∣ x ∣) + ∣ f (x) ∣ = (x - 1) + (1 - x) = 0 \neq = 1$ . Statement (C) is false.

The others hold: on $[1, 2]$ , $f (x) = x - 1 \geq 0$ , so $∣ f (x) ∣ = x - 1$ and $f (∣ x ∣) - ∣ f (x) ∣ = 0$ ; on $[0, 1]$ , $f (∣ x ∣) = ∣ x ∣ - 1$ .

✓ Check

x = \frac{1}{2}

f (\frac{1}{2}) = - \frac{1}{2}

, so

f (∣ x ∣) + ∣ f (x) ∣ = - \frac{1}{2} + \frac{1}{2} = 0

, not 1.

⚠ Common trap

(D) looks false if you expect the modulus to flip the sign, but on

[1, 2]

x - 1 \geq 0

, so

∣ f (x) ∣ = f (x)

and the difference is

0

Quick tip

On $[0, 1]$ : $f (∣ x ∣) = x - 1$ and $∣ f (x) ∣ = 1 - x$ , so the sum is 0, not 1

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Functions — Preparation

This question tests your understanding of Functions, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A function assigns exactly one output to each input in its domain. This topic covers the domain, codomain and range of a function, types of functions (one-one, onto, even and odd functions), and standard functions such as the identity, constant, polynomial, rational, modulus, signum, greatest integer, exponential and logarithmic functions, along with their graphs. It also covers the composition of functions, the inverse of a function, and finding the domain and range of a given expression. For the domain, look for values that make a denominator zero, the expression under a square root negative, or the argument of a logarithm non-positive, and exclude them. A function has an inverse only if it is both one-one and onto. Quick graph sketches help a great deal with questions on range and on the number of solutions of an equation. Practise the functions questions below, each with a step-by-step solution.

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