CAT Modulus functions MCQ - Practice Questions & Answers

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Modulus functions

Definition:

A modulus function (also known as the absolute value function) is defined as:

f(x) = |x|

Where:

$\mathrm{\left\{\begin{array}{l} x \text { if } x \geq 0 \\ -x \text { if } x<0 \end{array}\right.}$

The modulus function gives the "distance" of a number from zero on the real number line, without considering its sign.

Characteristics:

- The graph of |x| is a V-shaped curve that touches the x-axis at the origin.

- It's always non-negative.

Application of Even and Odd Functions in Modulus:

The modulus function is an even function because for all x :

|-x| = |x|

Foundation Building Questions:

Question 1:

Solve for x if |x - 3| = 5 .

Solution:

From the definition of the modulus, we have two cases:

1. x - 3 = 5 which gives x = 8

2. x - 3 = -5 which gives x = -2

Question 2:

Given $\mathrm{f(x)=\left|x^2-4 x+3\right|}$ , determine whether the function is even, odd, or neither.

Solution:

First, factorise the quadratic: $\mathrm{x^2-4 x+3=(x-3)(x-1)}$ .

For the function to be even, f(-x) should be equal to f(x).

Evaluate $\mathrm{f(-x):\left|-x^2+4 x-3\right|}$ which is not equal to f(x).

For the function to be odd, f(-x) should be equal to -f(x).

However, since the function's output is always non-negative due to the modulus, it can't be odd.

Thus, the function is neither even nor odd.

Tips and Tricks related to 'Modulus Functions':

1. Dual Nature:

- Always remember that a modulus equation can give rise to two separate equations, one for the positive and another for the negative value of the expression inside the modulus.

2. Graphical Representation:

- The graph of a modulus function can help visualise solutions. For |x|, it's a V-shape. When combined with other functions, the graph can have multiple "V" segments.

3. Interaction with Even/Odd Functions:

- If a function inside the modulus is even, then the modulus doesn't change its nature. However, if it's odd, the modulus can make it neither even nor odd, as seen in the example above.

4. Solving Equations:

- When solving |f(x)| = a, consider both f(x) = a and f(x) = -a.

Understanding the modulus function is crucial as it often appears in various complex equations in competitive exams. The key is to remember its basic definition and apply it systematically.