Functions, Domain, Range & Graphs | CAT & MBA CET Quant Vault

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DEFINITION

One-Line Definition

A Function $f: X \to Y$ is a mathematical machine that maps every input $x$ in Domain $X$ to exactly one output $y$ in Codomain $Y$. The set of actual outputs is the Range.

Golden Condition: Range $\subseteq$ Codomain (Range = Codomain if ONTO).

CORE INTUITION ⚡

Input Restrictions & Symmetry Model

Mastering Functions requires 3 mental models:

• Domain Exclusion: Exclude $x$ that causes zero denominators or negative square roots.
• Symmetry: Even functions reflect across Y-axis ($f(-x)=f(x)$); Odd functions reflect across Origin ($f(-x)=-f(x)$).
• Graph Shifts: $f(x-c)$ shifts RIGHT, $f(x)+c$ shifts UP, $|f(x)|$ flips negative Y parts UP.

Always check vertical line test: one input $x$ can NEVER have two outputs $y$!

💡 WHY THIS CONCEPT MATTERS & REAL-LIFE APPLICATIONS

Functions and Graphs form the backbone of Advanced Algebra, Optimization, and Calculus in CAT & MBA CET. Click below to explore connected Quant Vault topics:

Maxima & Minima Inequalities & Modulus Quadratic Equations Polynomials Logarithms

Where Is This Used in Real Life & Business?

📊 Demand & Supply Modeling ($P = f(Q)$)

📈 Machine Learning Neural Net Mappings

💰 Financial Revenue Optimization

🔄 Signal Processing Transformations

1 Definition of a Function & Vertical Line Test

A Function $f$ from a set $X$ (Domain) to a set $Y$ (Codomain), written as $f: X \to Y$, is a specific rule that assigns to every element $x \in X$ exactly one unique element $y \in Y$.

📌 Key Formal Properties:

1. Domain ($X$): The complete set of allowable input values $x$ for which $f(x)$ is defined.

2. Codomain ($Y$): The overall target set containing all possible outputs.

3. Range: The actual set of outputs produced: $\text{Range}(f) = \{ f(x) \mid x \in X \} \subseteq \text{Codomain}$.

📈 Vertical Line Test for Graphs

A graph in the $xy$-plane represents a function if and only if **no vertical line** intersects the graph at more than one point. If any vertical line cuts the curve at 2 or more points (e.g. a circle $x^2 + y^2 = r^2$), the relation is **NOT** a function!

Classification of Functions

1. One-to-One (Injective)

Distinct inputs produce distinct outputs: $f(x_1) = f(x_2) \implies x_1 = x_2$. Passes Horizontal Line Test.

2. Many-to-One

Two or more different inputs map to the same output. Example: $f(x) = x^2$ (since $f(2) = f(-2) = 4$).

3. Onto (Surjective)

Every element in the Codomain has at least one pre-image in the Domain ($\text{Range} = \text{Codomain}$).

4. Bijective (One-to-One & Onto)

A function that is both Injective and Surjective. **Bijectivity is the MANDATORY condition for an inverse function $f^{-1}(x)$ to exist!**

2 Rules for Finding Domain & Range

🔍 3 Golden Rules for Finding Domain

Rule 1 (Denominators): For $f(x) = \frac{g(x)}{h(x)}$, set $h(x) \ne 0$. Exclude roots of $h(x)$.
Rule 2 (Square Roots): For $f(x) = \sqrt{g(x)}$, set $g(x) \ge 0$. (Inside of even roots must be non-negative).
Rule 3 (Logarithms): For $f(x) = \log_b(g(x))$, set $g(x) > 0$, base $b > 0$, and $b \ne 1$.

🎯 Strategies for Finding Range

Method A

Express $x$ in terms of $y$: Set $y = f(x)$, solve for $x = g(y)$, and find the domain of $g(y)$ with respect to $y$.

Method B

Quadratic Bounds: For $a x^2 + b x + c$: if $a > 0$, Range is $\left[-\frac{D}{4a}, \infty\right)$. If $a < 0$, Range is $\left(-\infty, -\frac{D}{4a}\right]$.

Method C

AM-GM Bounds: Use $x + \frac{1}{x} \ge 2$ for positive $x$ to establish lower/upper bounds.

3 Even vs Odd Functions & Symmetry Rules

EVEN FUNCTION

$$f(-x) = f(x)$$

• Symmetry: Symmetric about the Y-axis.
• Examples: $f(x) = x^2, \cos(x), |x|, x^4 + 5$.
• Property: Replacing $x$ with $-x$ leaves equation unchanged.

ODD FUNCTION

$$f(-x) = -f(x)$$

• Symmetry: Symmetric about the Origin $(0,0)$.
• Examples: $f(x) = x^3, \sin(x), \frac{1}{x}, x^5 - 3x$.
• Property: $f(0) = 0$ for any odd function defined at $x=0$.

4 Composite Functions $f(g(x))$ & Inverse Functions $f^{-1}(x)$

🔄 Composite Function: $f(g(x))$

Given functions $f$ and $g$, the composite function $(f \circ g)(x) = f(g(x))$ applies $g$ first, then applies $f$ to the result.

• Non-Commutative: In general, $f(g(x)) \ne g(f(x))$.

• Associative: $f(g(h(x))) = (f \circ g \circ h)(x)$.

• Domain of $f(g(x))$: $x$ must be in Domain of $g$, AND $g(x)$ must be in Domain of $f$.

↩️ Inverse Function: $f^{-1}(x)$

If $f: X \to Y$ is a **Bijective** function, its inverse $f^{-1}: Y \to X$ satisfies:

$$f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x$$

• Graph Reflection: The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ across the line $y = x$.

• Domain & Range Swap: $\text{Domain}(f^{-1}) = \text{Range}(f)$ and $\text{Range}(f^{-1}) = \text{Domain}(f)$.

5 Graph Transformation Rules Cheat Sheet

Transformation	Formula	Geometric Effect
Vertical Shift Up	$y = f(x) + c$	Shift entire graph UP by $c$ units.
Vertical Shift Down	$y = f(x) - c$	Shift entire graph DOWN by $c$ units.
Horizontal Shift Right	$y = f(x - c)$	Shift entire graph RIGHT by $c$ units.
Horizontal Shift Left	$y = f(x + c)$	Shift entire graph LEFT by $c$ units.
Reflection across X-axis	$y = -f(x)$	Flip graph vertically across X-axis.
Reflection across Y-axis	$y = f(-x)$	Flip graph horizontally across Y-axis.
Modulus of Output	$y = \|f(x)\|$	Reflect any part below X-axis upward above X-axis.
Modulus of Input	$y = f(\|x\|)$	Delete left side ($x < 0$), mirror right side ($x > 0$) onto left.

⚡ Interactive Functions & Domain Calculators

1. Composite Evaluator

Interactive

Computes $(f \circ g)(x)$ and $(g \circ f)(x)$ for $f(x) = ax + b$ and $g(x) = cx^2 + d$.

f(x): slope a

f(x): intercept b

g(x): coeff c

g(x): constant d

Input x value

2. Symmetry & Domain Checker

Interactive

Tests symmetry $f(-x) = \pm f(x)$ and domain bounds for polynomial $f(x) = ax^3 + bx^2 + cx + d$.

Coeff a (x³)

Coeff b (x²)

Coeff c (x)

Constant d

⚠️ COMMON MISTAKES TO AVOID

❌ Mistake 1: Assuming $f(g(x)) = g(f(x))$

Function composition is non-commutative! $f(g(x))$ applies $g$ first, whereas $g(f(x))$ applies $f$ first. Always evaluate inner functions from right to left.

❌ Mistake 2: Forgetting to check domain of inner function in $f(g(x))$

Even if $f(g(x))$ simplifies algebraically to a simple polynomial, any $x$ value that makes $g(x)$ undefined is strictly excluded from the composite domain.

❌ Mistake 3: Attempting to invert non-bijective functions

$f(x) = x^2$ over $\mathbb{R}$ does NOT have an inverse because it fails the horizontal line test ($f(2) = f(-2) = 4$). You must restrict the domain to $x \ge 0$ first!

🚀 CAT & MBA CET FUNCTIONAL EQUATION SHORTCUTS

⚡ Shortcut 1: Multiplicative-to-Additive Functional Identitiy

If $f(x \cdot y) = f(x) + f(y)$ for all positive real numbers, then:

$$f(x) = k \cdot \ln(x) \quad \text{or} \quad f(x) = k \cdot \log_b(x)$$

⚡ Shortcut 2: Additive-to-Multiplicative Functional Identity

If $f(x + y) = f(x) \cdot f(y)$ for all real numbers, then:

$$f(x) = a^x \quad (a > 0)$$

⚡ Shortcut 3: Polynomial Identity $f(x) + f(1/x) = f(x) \cdot f(1/x)$

For any polynomial $f(x)$ satisfying $f(x) + f\left(\frac{1}{x}\right) = f(x) \cdot f\left(\frac{1}{x}\right)$:

$$f(x) = 1 \pm x^n$$

🎯 PRACTICE QUESTIONS (DIFFICULTY LEVEL-WISE)

EASY • LEVEL 0 DOMAIN DETERMINATION

Find the domain of the real-valued function $f(x) = \frac{1}{\sqrt{x^2 - 16}}$.

MODERATE • LEVEL 1 INVERSE FUNCTION EVALUATION

If $f(x) = \frac{3x + 2}{2x - 5}$, find $f^{-1}(4)$.

HARD • LEVEL 2 CAT FUNCTIONAL EQUATION

A polynomial function $f(x)$ satisfies $f(x) + f(1/x) = f(x) \cdot f(1/x)$ for all $x \ne 0$. If $f(3) = 28$, find $f(4)$.

❓ FREQUENTLY ASKED QUESTIONS

Q: What is the difference between Codomain and Range?

Codomain is the complete target set $Y$ specified in $f: X \to Y$. Range is the subset of $Y$ containing only those values actually hit by $f(x)$. Range is equal to Codomain if and only if the function is Onto (Surjective).

Q: Can a function be neither Even nor Odd?

Yes! Most functions (e.g. $f(x) = x^2 + x$) are neither even nor odd because $f(-x) \ne f(x)$ and $f(-x) \ne -f(x)$. The only function that is both even and odd is the zero function $f(x) = 0$.

Q: How do you graph y = |f(x)| versus y = f(|x|)?

For $y = |f(x)|$, take the graph of $y = f(x)$ and flip any portion lying below the X-axis upward. For $y = f(|x|)$, delete the entire graph on the left of the Y-axis ($x < 0$) and replace it with the Y-axis reflection of the right half ($x > 0$).

Q: Why must a function be Bijective to have an inverse f^-1(x)?

If a function is not One-to-One, the inverse mapping would map a single input to multiple outputs (failing the vertical line test). If it is not Onto, some elements in the inverse domain would have no output. Thus, bijectivity ensures $f^{-1}(x)$ is a valid function.

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