Basic Concepts of Functions and Graphs for CAT Exams

Basic Concepts of Functions and Graphs for CAT Exams In CAT exams, there are certain algebraic questions that require higher level concepts. However, these questions can also be solved using alternate approaches from graphs. Below are a few challenging questions that would be difficult to solve using algebraic concepts alone:

1. Find the number of real solutions of x for the equation 2^x = x^2.
2. Find the number of integral solutions to the equation |x-1| + |x+3| + |x-5| < 20. 3. How many integral pairs (x, y) satisfy |x| + |y| = 4? 4. Find the minimum value of y if y = |1 – |x+3||. While these questions can be solved using algebra, having a basic understanding of the graphs of special functions and their transformations can make them easier to solve.

This article will focus on these concepts, while the next article will explain how to use them to solve the above types of questions. Let’s start by understanding some special functions that frequently appear in CAT exams:

1. Identity function/ graph of f(x) = x: The identity function is represented by a straight line passing through the origin and inclined at a 45° angle with the x-axis.

2. Graph of f(x) = x^2: This is a parabola and is an even function, meaning it is symmetrical about the y-axis.

3. Graph of f(x) = x^3: This is a cube function and is an odd function, meaning it is symmetrical about the origin or opposite quadrants.

By analyzing and generalizing the above graphs, we can conclude that: – The graph of f(x) = x^(2n) (where n is a natural number) is always an even function, symmetrical about the y-axis. – The graph of f(x) = x^(2n-1) (where n is a natural number) is always an odd function, symmetrical about the origin or opposite quadrants. Rational expressions also play a role in CAT exams. These functions are obtained by dividing one polynomial by another. Let’s look at some basic rational functions and their graphs:

1. Graph of f(x) = 1/x: This is a reciprocal function or a rectangular hyperbola. It is also an odd function, symmetrical about the opposite quadrant.

2. Graph of f(x) = 1/x^2: This is an even function, symmetrical about the y-axis. It is important to note that the domain of this function is all real values except 0.

Piecewise functions are another category of functions that appear in CAT exams. The focus is mainly on the absolute value function:

1. Absolute value function (or modulus function): This function is defined as y = |x| and has two cases: – For x ≥ 0, y = x. – For x < 0, y = -x.

The function is symmetrical about the y-axis. Exponential and logarithmic functions are also important concepts:

1. Exponential function: The function f(x) = a^x (where a ≠ 0) and x is a real number is called an exponential function. Depending on the value of a, the function can either be increasing or decreasing.

2. Logarithmic function: The logarithmic function is the inverse of the exponential function. It is defined as f(x) = log_a(x), where x and a are positive real numbers and a ≠ 0. These are the basic functions and their graphs that frequently appear in CAT exams.

Understanding the transformations of these graphs can help simplify complex questions. Here are some transformations:

1. Transformation of function f(x) to f(x) ± a: – Add or subtract ‘a’ to shift the graph vertically.

2. Transformation of function f(x) to f(x ± a): – Add or subtract ‘a’ to shift the graph horizontally.

3. Transformation of function f(x) to -f(x): – Reflect the graph in the x-axis.

4. Transformation of function f(x) to f(-x): – Reflect the graph in the y-axis.

5. Transformation of function f(x) to -f(-x): – Reflect the graph in both the x-axis and y-axis.

6. Transformation of function f(x) to |f(x)|: – Take the mirror image (in x-axis) of the part of the graph that lies below the x-axis.

7. Transformation of function f(x) to f(|x|): – Neglect the graph for x < 0 and take the mirror image of the remaining part of the graph about the y-axis.

These transformations are sufficient to handle most questions in CAT exams and help you to understand the Basic Concepts of Functions and Graphs for CAT Exams. In the next article, we will discuss how to use these concepts to solve the types of questions mentioned earlier.