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Functions (Advanced) revision notes

Every idea in the Functions (Advanced) chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 15 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Domain of combined expressions

Logs with variable base and nested logs
4× asked, last 2025
log⁡g(x)f(x): f>0, g>0, g≠1,log⁡au>k  ⟺  u>ak  (a>1)\log_{g(x)}f(x):\ f>0,\ g>0,\ g\ne1,\qquad \log_a u>k\iff u>a^k\ \ (a>1)
For 0<a<10<a<1 the inequality flips: 0<u<ak0<u<a^k. Every argument and base in the chain needs its own condition.
  • Quick method. Work from the outside in: log⁡4log⁡3log⁡7u\log_4\log_3\log_7u needs log⁡3log⁡7u>0\log_3\log_7u>0, so log⁡7u>1\log_7u>1, so u>7u>7. Do not drop u>0u>0 when an outer log gives log⁡4u<1\log_4u<1: it means 0<u<40<u<4.
Inverse sine or cosine of a rational function
7× asked, last 2025
−1≤pq≤1  ⟺  p−qq≤0  and  p+qq≥0-1\le\frac{p}{q}\le1\iff\frac{p-q}{q}\le0\ \text{ and }\ \frac{p+q}{q}\ge0
Same condition for sin⁡−1\sin^{-1} and cos⁡−1\cos^{-1}. Solve each part by a sign chart; q=0q=0 is always excluded.
times (x+2)(x+2)x−3x+2≤0\frac{x-3}{x+2}\le0−2-23++−-++sign of x−3x+2\frac{x-3}{x+2}
2x−1x+2≤1\frac{2x-1}{x+2}\le1: cross-multiplying keeps x<−2x<-2 wrongly; the sign chart gives −2<x≤3-2<x\le3
  • Combined expressions: find each part's domain from the original expression, before simplifying, then intersect (rules on Relations & Functions (Basic)).
  • Not cross-multiplying by x+2x+2. Its sign is unknown, so the inequality may flip. Move everything to one side: (2x−1)/(x+2)≤1(2x-1)/(x+2)\le1 becomes (x−3)/(x+2)≤0(x-3)/(x+2)\le0, so −2<x≤3-2<x\le3.

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Practise Functions (Advanced)

13 more concepts in 4 topics

Range of a function; Composite and inverse functions; Even, odd, periodic functions and graphs; and Functional equations. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Range of a function

  • Quadratic-in-y (discriminant) method2×
  • Monotonicity, end values and bijections2×
  • Bounded expressions and AM-GM4×

Composite and inverse functions

  • Inverse function and its graph5×
  • Inverse of a bijection: values and slopes4×
  • Also remember1

Even, odd, periodic functions and graphs

  • Even and odd functions1×
  • Periodic functions
  • Graph transformations and counting roots

Functional equations

  • Additive (Cauchy) equation8×
  • Exponential equation f(x+y) = f(x)f(y)8×
  • Power, log and cosine types5×
  • Systems in f(x) and f(1/x)7×
  • Pairing sums: f(x) + f(t − x)4×
  • Also remember1

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