Types — constant, linear, quadratic, polynomial, rational, radical, piecewise

Overview

Functions are classified by their algebraic form and behavior. Understanding function types helps you predict their graphs, choose appropriate solution methods, and recognize patterns across mathematics. Each type has a signature structure that determines its domain, range, continuity, and asymptotic behavior.

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1. Constant Functions

Derivation from scratch:

• A function maps input $x$ to output $y$
• If the mapping rule is "ignore $x$, always return $c$", then $f(x) = c$
• Why? No matter what $x$ you plug in, the equation has no $x$ term, so output can't depend on input

Properties:

• Domain: $\mathbb{R}$ (all real numbers)
• Range: $\{c\}$ (single value)
• Graph: horizontal line at $y = c$
• Slope: $0$ (no change)

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2. Linear Functions

Derivation from first principles:

1. Start with the idea: "change in output is proportional to change in input" $\frac{\Delta y}{\Delta x} = m \text{ (constant)}$
2. Integrate this rate: $\Delta y = m \cdot \Delta x$
3. If we start at point $(0, b)$, then after moving $x$ units horizontally: $y - b = m \cdot (x - 0)$
4. Solve for $y$: $y = mx + b$✓

Why this form?

• $m$: How steep? Positive = rising, negative = falling, zero = horizontal
• $b$: Where does it cross the $y$-axis?

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3. Quadratic Functions

Derivation from scratch:

• Suppose we have constant acceleration $2a$
• Velocity: $v(x) = 2ax + b$ (linear)
• Position (integrate velocity): $f(x) = \int (2ax + b) dx = ax^2 + bx + c$
• Why $x^2$? Accumulating a linearly changing rate gives a quadratic

Standard Forms:

1. Standard form: $f(x) = ax^2 + bx + c$

   • $c$: $y$-intercept

2. Vertex form: $f(x) = a(x - h)^2 + k$

   • Vertex at $(h, k)$
   • How to convert? Complete the square

3. Factored form: $f(x) = a(x - r_1)(x - r_2)$

   • Roots/zeros at $x = r_1, r_2$

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4. Polynomial Functions

Key Properties by Degree:

• Degree 0: Constant ($a_0$)
• Degree 1: Linear ($ax + b$)
• Degree 2: Quadratic ($ax^2 + bx + c$)
• Degree 3: Cubic ($ax^3 + bx^2 + cx + d$)
• Degree $n$: Up to $n$ roots, $n-1$ turning points

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5. Rational Functions

Domain: All real numbers except where $q(x) = 0$.

Types of Asymptotes:

1. Vertical asymptote at $x = a$ if $q(a) = 0$ and $p(a) \neq 0$

   • Function approaches $\pm\infty$ as $x \to a$

2. Horizontal asymptote $y = L$ if $\lim_{x \to \infty} f(x) = L$

   • Determined by degrees of $p$ and $q$:
     • deg$(p) <$ deg$(q)$: $y = 0$
     • deg$(p) =$ deg$(q)$: $y = \frac{a_n}{b_n}$ (ratio of leading coefficients)
     • deg$(p) >$ deg$(q)$: no horizontal asymptote (may have oblique)

3. Oblique asymptote: if deg$(p) =$ deg$(q) + 1$, perform polynomial division

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6. Radical Functions

Domain restrictions:

• Even roots ($\sqrt{x}$, $\sqrt[4]{x}$, ...): radicand must be $\geq 0$ (no negative under even root in real numbers)
• Odd roots ($\sqrt[3]{x}$, $\sqrt[5]{x}$, ...): any real number OK

Derivation of domain:

• For $f(x) = \sqrt{g(x)}$ to be real, we need $g(x) \geq 0$
• Why? $\sqrt{a}$ is defined as the number $b \geq 0$ such that $b^2 = a$. No real $b$ satisfies $b^2 < 0$.

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7. Piecewise Functions

Important concepts:

• Continuity at boundaries: Does $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$?
• Evaluating: Find which piece applies, then use that formula
• Graphing: Graph each piece on its domain, watch for jumps/gaps

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Summary Table

| Type | General Form | Key Feature | Domain | |------|--------------|-----| | Constant | $f(x) = c$ | Horizontal line | $\mathbb{R}$ | | Linear | $f(x) = mx + b$ | Straight line, constant slope | $\mathbb{R}$ | | Quadratic | $f(x) = ax^2 + bx + c$ | Parabola | $\mathbb{R}$ | | Polynomial | $f(x) = a_n x^n + \cdots + a_0$ | Smooth curve, degree $n$ | $\mathbb{R}$ | | Rational | $f(x) = \frac{p(x)}{q(x)}$ | Asymptotes | $q(x) \neq 0$ | | Radical | $f(x) = \sqrt[n]{g(x)}$ | Root function | $g(x) \geq 0$ (even $n$) | | Piecewise | Multiple formulas | Different rules on intervals | Varies by piece |

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Recall Feynman Technique: Explain to a 12-Year-Old

Imagine you have a machine that takes number and gives you back another number. Different machines have different rules: Constant machine: No matter what number you put in, it always spits out the same number. Like a broken vending machine that only gives you chocolate, even if you press "chips."

Linear machine: For every step you go forward on the input, the output goes up (or down) by the same amount. Like climbing stairs—each step up takes you the same height.

Quadratic machine: The output changes faster and faster (or slower and slower). Like a ball you throw up—it slows down, stops, then speeds up coming back down. Makes a U-shape.

Rational machine: Divides one polynomial by another. Sometimes it goes crazy and shoots to infinity (vertical asymptote) where the bottom is zero. Like trying to divide pizza among zero people—impossible!

Radical machine: Takes the opposite of squaring. If you squared something to get 9, the radical machine gives you back 3. But it's picky—you can't square root a negative (in regular numbers).

Piecewise machine: Has multiple personalities! It uses one rule when the input is small, a different rule when it's big. Like ticket prices: kids pay less, adults pay more.

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Connections

• M02.01 Function Definition — understanding what a function is before classifying types
• M02.03 Domain and Range — each type has characteristic domain/range restrictions
• M02.04 Function Transformations — shifting, stretching, reflecting any of these types
• M02.06 Inverse Functions — some types (like quadratics) need domain restrictions to be invertible
• M03.01 Limits — needed to define asymptotes in rational functions rigorously
• M03.02 Continuity — piecewise functions often have continuity issues at boundaries
• M04.01 Derivatives — each function type has characteristic derivative formulas
• M05.01 Integration — integrating polynomials vs. rational functions require different techniques

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#flashcards/maths

What is a constant function? :: A function $f(x) = c$ where the output is always the same value $c$ regardless of input.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho beta, functions ke types samajhna bilkul aisa hai jaise alag-alag tools ka dabba pehchaanna. Har function ka apna ek "signature structure" hota hai jo bata deta hai ki uska graph kaisa dikhega aur woh kaise behave karega. Jaise constant function ka matlab hai output hamesha same rehta hai chahe aap koi bhi input daalo, jaise light ki speed jo kabhi change nahi hoti. Wahi linear function mein "constant rate of change" hota hai — matlab har baar x ek unit badhta hai toh y utna hi (slope m jitna) badhta hai, seedhi line ki tarah, jaise stairs chadhna jahan har step ki height barabar ho. Aur quadratic mein "acceleration" aata hai — yahan rate of change khud badalta rehta hai, isliye graph ek parabola banta hai jaise upar phenki gayi ball ka path.

Ab ye samajhna kyun zaroori hai? Kyunki jab aap ek function ka type pehchaan lete ho, tab aap uske baare mein bina calculate kiye hi bahut kuch predict kar sakte ho — jaise domain, range, graph ka shape, aur solve karne ka sahi tarika. Jaise ki linear function mein slope (m) hi decide karta hai ki function badh raha hai ya ghat raha hai, na ki intercept (b). Bahut students yahan galti karte hain — woh sochte hain agar b positive hai toh function badh raha hoga, lekin actually agar m negative hai toh function decrease hi karega, b sirf starting height batata hai.

Ye classification isliye powerful hai kyunki poori maths mein aapko yeh patterns baar-baar milenge — cost calculations mein linear, motion mein quadratic, aur physics constants mein constant functions. Ek baar ye intuition mazboot ho gayi na, toh aap naye problems dekhte hi samajh jaoge ki kaunsa approach lagana hai, jisse aapka time bhi bachega aur confidence bhi badhega. Isliye har type ka core structure dil se samajh lo, ratta mat maro!