CalculusHigh Yield (2 Qs in JEE)

Functions & Relations Formulas

Domain, range, injective (one-one), surjective (onto), composite functions, inverse functions, and functional equations.

8 Formulas•552+ Questions in Bank

Core Formulas & Mathematical Identities

\sqrt{f(x)} \implies f(x) \ge 0, \quad \ln(f(x)) \implies f(x) > 0, \quad \frac{1}{f(x)} \implies f(x) \ne 0

Fundamental algebraic domain restrictions for real-valued functions.

Examiner Tip: Domain of sin⁻¹(f(x)) and cos⁻¹(f(x)) requires -1 ≤ f(x) ≤ 1.

f(x_1) = f(x_2) \implies x_1 = x_2 \quad \text{or} \quad f'(x) > 0 \; \forall x \text{ (Strictly Monotonic)}

Horizontal line test: every horizontal line intersects the graph at most once.

Examiner Tip: Strictly increasing or strictly decreasing continuous functions on an interval are always one-one.

\text{Range}(f) = \text{Codomain}(f)

Every element in the codomain has at least one pre-image in the domain.

Examiner Tip: Odd degree polynomial functions with codomain R are always onto R.

(f \circ g)(x) = f(g(x)), \quad f(f^{-1}(x)) = x, \quad (f \circ g)^{-1} = g^{-1} \circ f^{-1}

Function composition and reversal law for inverse functions.

Examiner Tip: Graph of f⁻¹(x) is the mirror reflection of f(x) across the line y = x.

f(xy) = f(x) + f(y) \implies f(x) = k\ln x; \quad f(x+y) = f(x)f(y) \implies f(x) = a^x; \quad f(x) + f(1/x) = f(x)f(1/x) \implies f(x) = 1 \pm x^n

Classic functional identities frequently tested in JEE Advanced.

Examiner Tip: Substitute x = 0, 1, or -1 to find base constants.

\log_a(xy) = \log_a x + \log_a y, \quad \log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y, \quad \log_b a = \frac{\log_c a}{\log_c b}, \quad a^{\log_b c} = c^{\log_b a}, \quad \log_{b^\beta}(a^\alpha) = \frac{\alpha}{\beta}\log_b a

Exhaustive logarithmic laws and base-change properties for solving exponential and log expressions.

Examiner Tip: Always ensure base a > 0, a ≠ 1 and argument x > 0 before manipulating equations.

\text{For } \log_a x > \log_a y: \begin{cases} x > y > 0 & \text{if } a > 1 \\ 0 < x < y & \text{if } 0 < a < 1 \end{cases}; \quad x = [x] + \{x\}, \quad \log_{10} N = \text{Char} + \text{Mantissa}

Inequality reversal depending on base a, and number of digits in N is equal to (Characteristic + 1).

Examiner Tip: Number of leading zeroes before first significant digit after decimal in N (0 < N < 1) is |Characteristic| - 1.

f(-x) = f(x) \iff \text{Even (Symmetric about y-axis)}; \quad f(-x) = -f(x) \iff \text{Odd (Symmetric about origin)}; \quad f(x+T) = f(x) \iff \text{Periodic with Period } T

Symmetry and periodicity properties. Derivative of an even function is odd, and vice-versa.

Examiner Tip: Period of |sin x|, |cos x|, |tan x| is π. Period of sin(ax + b) is 2π/|a|.

Frequently tested misconceptions in Functions & Relations

•Forgetting that domain of √(f(x)) requires f(x) ≥ 0, but for 1/√(f(x)) it requires f(x) > 0.
•Invertible function must be BOTH one-one and onto (bijective).