Step-by-Step Strategies for Functions and Inequalities in CAT Quant Section

The basic strategy to solve questions on Functions and Inequalities in CAT Quant Section is to first check the domain. For functions always look for symmetry form while for inequalities use the number line test.

Table of Contents

Step-by-Step Strategy for Functions
Step-by-Step Strategy for Inequalities
Tricks to Solve Functions in CAT
Tricks to Solve Inequalities in CAT
Faqs

In the CAT exam, usually questions from Functions and Inequalities are usually integrated with topics like quadratic equations, modulus, relations, differentiation. With respect to questions in CAT Functions and Inequalities from Quant Section it is important to start with domain analysis and thereby classify the function whether it is linear, quadratic or modulus. Moreover for the inequalities section it is important to express the inequality in the standard form and isolate the variable on the one side of the equation. In this article we have systematically highlighted on how to solve these questions correctly in the CAT exam.

Also Read: Analyzing Your CAT Mock Test

Step-by-Step Strategy for Functions

Aspirants can follow the steps to solve the questions on functions.

Steps	Details
Understand the Question	Take note that the domain of the function is the valid input while the range is the possible output
Check Domain of Function	In order to check domain restrict the value to Denominators (≠ 0) Square roots (≥ 0) Logarithms (> 0) Absolute values (case-split if needed)
Next Find Range	To find range follow the transformation method by using Quadratic, Rational and Composition techniques
Use Graphical Representation	Rough sketches help with symmetry (odd/even functions), increasing/decreasing behavior, and boundary values.

For Functions, candidates must take note that

Use Substitution Method for options
f(x+y) = f(x)+f(y) → linear function
Modulus Function - Split two cases

Also Read: Effective Guessing Strategies for CAT Verbal Ability

Step-by-Step Strategy for Inequalities

Aspirants can follow the steps to solve the questions on inequalities

Steps	Details
Simplify Expression	Factorise the quadratic equation. Bring the values to one side
Use Number Line Method	Plot critical points on a line. Test sign in each interval. Decide open/closed interval depending on ≥, ≤
Absolute Value	Split into cases based on sign
Exponential / Logarithmic	If base > 1 → inequality direction unchanged. If 0 < base < 1 → inequality flips.

For Inequalities, candidates must take note that

Use symmetry (e.g., |x| ≥ k always gives two branches).
Eliminate wrong options quickly with test values.
For equation ax² + bx + c > 0 → solution lies outside roots if a > 0.
For equation ax² + bx + c < 0 → solution lies between roots if a > 0.

Also Read: Caselets vs Graphs in CAT DILR and How to Choose

Tricks to Solve Functions in CAT

Applicants can follow the methods to solve Functions in CAT exam.

Method	Detailed Steps
Substitution Method	When asked to find f(f(x)) or f(g(x)) substitute step by step Tip: If f(x) = x2+1 then f(f(x)) = f(x2+1) = (x2+1)2+1.
Use Graph Visualization	Replace f(x)=∣g(x)∣ with cases If g(x) ≥ 0 then f(x) = g(x) If g(x) < 0 then f(x) = - g(x)
Piecewise Functions	For modulus or floor functions, break into intervals Example: f(x) = ∣x−2∣ Case 1 - x ≥ 2 → f(x) = x−2 Case 2 - x < 2 → f(x) = - (x - 2) = 2 - x

Method

Detailed Steps

Substitution Method

When asked to find f(f(x)) or f(g(x)) substitute step by step

Tip:

If f(x) = x2+1 then f(f(x)) = f(x2+1) = (x2+1)2+1.

Use Graph Visualization

Replace f(x)=∣g(x)∣ with cases

If g(x) ≥ 0 then f(x) = g(x)
If g(x) < 0 then f(x) = - g(x)

Piecewise Functions

For modulus or floor functions, break into intervals

Example: f(x) = ∣x−2∣

Case 1 - x ≥ 2 → f(x) = x−2

Case 2 - x < 2 → f(x) = - (x - 2) = 2 - x

Tricks to Solve Inequalities in CAT

Applicants can follow the methods to solve Functions in CAT exam.

Methods	Tricks
Tricks for Linear Inequalities	Solve like equations, but flip the inequality sign when multiplying/dividing by a negative number. For example- -2x > 6 ⟹ x < –3
Tricks for Quadratic Inequalities	Factorize the quadratic (if possible). Find the critical points (roots) If coefficient of x² > 0 → parabola opens upward If the coefficient of x² < 0 → parabola opens downward.
Tricks for Rational Inequalities	Factorize numerator & denominator separately. Mark critical points (where numerator = 0 or denominator = 0). Draw number line → test signs of each interval. Exclude points where denominator = 0.

Methods

Tricks

Tricks for Linear Inequalities

Solve like equations, but flip the inequality sign when multiplying/dividing by a negative number.

For example-

-2x > 6 ⟹ x < –3

Tricks for Quadratic Inequalities

Factorize the quadratic (if possible).
Find the critical points (roots)
If coefficient of x² > 0 → parabola opens upward
If the coefficient of x² < 0 → parabola opens downward.

Tricks for Rational Inequalities

Factorize numerator & denominator separately.
Mark critical points (where numerator = 0 or denominator = 0).
Draw number line → test signs of each interval.
Exclude points where denominator = 0.

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FAQs

What’s a step-by-step trick for solving rational inequalities (fractions)?

Identify where numerator = 0 and denominator = 0.
Mark these as critical points.
Test intervals between them.
Remember: denominator ≠ 0, so exclude those points.

How should I approach modulus-based function problems step-by-step?

Remove modulus by considering different cases (+ and –).
Solve each case separately.
Ensure the solution satisfies the condition of that case.
Combine valid solutions.

Should I always check boundary points in inequalities?

Yes. Boundary points (where the expression = 0 or undefined) determine the intervals where the inequality flips. Always test if they should be included (≤, ≥) or excluded (<, >).

What is the first step in solving function-based problems in CAT?

The first step is to understand the function’s definition and domain. Always check whether the function is polynomial, modulus-based, piecewise, or composite. This helps decide the solving strategy.

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