Important Formulas: Functions | Quantitative for GMAT PDF Download

Introduction

Functions are fundamental in modeling real-life relationships in business, economics, and analytics. For example, a profit function can relate the quantity of goods sold to total profit, aiding in decision-making and forecasting. Understanding functions is crucial for data analysis, optimization, and interpreting business scenarios.

What is a Function?

A function is a clear relationship between two sets:

• Domain (X): The set of all possible input values.

• Codomain (Y): The set of all possible output values.

A function f: X → Y assigns each element in X to exactly one element in Y.

• Independent variable (x): Input value.

• Dependent variable (y = f(x)): Output value.

Basic Methods of Representing Functions

1. Analytical Representation

A function is expressed using a formula:

• Single formula: y = 3x²

• Piecewise formula:
  

2. Graphical Representation
A function is plotted on the x-y plane, where each point (x, y) corresponds to (x, f(x)).

• Example: Graph of f(x) = x² for X = {-3, -1, 1, 3}.
  
  

3. Tabular Representation

A table lists x and corresponding y = f(x) values.

Key Formulas & Properties for Functions

1. Even Functions

2. Odd Functions

3. Neither Even Nor Odd

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4. Identity Function

• Defined as f(x) = x.

• Graph is a straight line at 45° passing through the origin.

5. Absolute Value Function

6. Exponential Function

7. Logarithmic Function

8. Greatest Integer (Floor) Function

9. Function Transformations (Critical for GMAT)

Vertical Shifts

• $f\left(x\right)+c$f(x)+c → Up by $c$c ✅

• $f\left(x\right)-c$f(x)−c → Down by $c$c ✅

Horizontal Shifts

• $f(x+c)$f(x+c) → Left by $c$c ✅

• $f(x-c)$f(x−c) → Right by $c$c ✅

Reflections

• $-f\left(x\right)$−f(x) → Reflects over x-axis ✅

• $f(-x)$f(−x) → Reflects over y-axis ✅

Absolute Value Transformation

• $\mid f\left(x\right)\mid$∣f(x)∣ → Negative parts flip above x-axis ✅

10. Quadratic Functions