Quant · Algebra

Functions and Graphs

3 sub-topics · 25 lessons · 77 questions · 15 PYQs

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Functions11 lessons · 39 questions · 6 PYQs

Graphs and Behaviour9 lessons · 24 questions · 7 PYQs

Relations5 lessons · 14 questions · 2 PYQs

Formula sheet

Inverse of a linear fractional function
$y=\dfrac{ax+b}{cx+d}\iff x=\dfrac{dy-b}{a-cy}$
Max with zero
$\max(a,0)=\begin{cases}a,&a\ge0\\0,&a<0\end{cases}$
Log forms and proportionality
$y=a+b\ln x\iff e^{y}=e^{a}x^{b};\qquad \ln y=a+b\ln x\iff y=e^{a}x^{b}$
Substitution in a functional equation
$f\left(x+\tfrac1x\right)=x^{2}+\tfrac{1}{x^{2}}\Rightarrow f(t)=t^{2}-2$
Maximum of the minimum of two lines
$\max_{x}\min(px+q,\,r-sx)\ \text{at}\ px+q=r-sx\quad(p,s>0)$
Domain rules
$\sqrt{g(x)}:\ g(x)\ge0,\qquad \dfrac{1}{g(x)}:\ g(x)\ne0,\qquad \log g(x):\ g(x)>0$
Size of a Cartesian product
$n(A\times B)=n(A)\,n(B),\qquad (A\times B)\cap(B\times A)=(A\cap B)\times(A\cap B)$
Number of onto functions
$\#\text{onto }A\to B=\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}(n-j)^{m}$