Operations on functions:

PYQ-2023-Function-Q2, PYQ-2023-Function-Q3, PYQ-2023-Function-Q4, PYQ-2023-Function-Q6, PYQ-2023-Function-Q7, PYQ-2023-Function-Q8, PYQ-2023-Function-Q11, PYQ-2023-Function-Q12, PYQ-2023-Function-Q14, PYQ-2023-Function-Q15, PYQ-2023-Function-Q18, PYQ-2023-Function-Q19, PYQ-2023-Function-Q20, PYQ-2023-Hyperbola-Q2

• $ (f+g)(x)=f(x)+g(x)$

• $ (f-g)(x)=f(x)-g(x)$

• $ (f.g) (x)=f(x) \cdot g(x)$

• $ (\frac{f}{g})(\mathrm{x})=\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})} ; \mathrm{g}(\mathrm{x}) \neq 0$

• $ (\mathrm{kf})(\mathrm{x})=\mathrm{kf}(\mathrm{x})$

Some special functions:

• if $f(x+y)=f(x)+f(y)$, then $f(x)=k x$

• if $f(x y)=f(x)+f(y)$, then $f(x)=\log x$

• if $f(x+y)=f(x) \cdot f(y)$, then $f(x)=e^x$

• if $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$, then $f(x)=x^n \pm 1$

Important Formulae:

PYQ-2023-Function-Q8, PYQ-2023-Function-Q13, PYQ-2023-Function-Q16, PYQ-2023-AOD-Q1

• The number of functions from a finite set $A$ into a finite set $B$ is $[n(B)]^{n(A)}$

• The number of one-one functions that can be defined from a set A into a finite set B is $ ^n(B)P_{n(A)} = \frac{n(B)!}{(n(B) - n(A))!} $; if $n(B) \geq n(A)$, otherwise is $0$

• The number of onto functions, that can be defined from a finite set A, containing n elements onto a finite set B, containing 2 elements = $2^n– 2$

• The number of onto functions from A to B where, $o(A) = m, o(B) = n $ and $m \geq n$ is $\sum_{\mathrm{r}=1}^{\mathrm{n}}(-1)^{\mathrm{n}-\mathrm{r~}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \mathrm{r}^{\mathrm{m}}$

• The number of bijections from a finite set A onto a finite set B is n(A)!; if n(A) = n(B) 0 ; otherwise

• If $o(A \cap B)=n$ then $\mathrm{o}[(\mathrm{A} \times \mathrm{B}) \cap(\mathrm{B} \times \mathrm{A})]=\mathrm{n}^2 $

• If any line parallel to $\mathrm{X}$-axis, cuts the graph of the function almost one point, then function is one-one.

• If there is even a single line parallel to $\mathrm{X}$-axis, cuts the graph of the function atleast two points, then function is many-one.

Domain and Range:

PYQ-2023-Function-Q9, PYQ-2023-Function-Q15, PYQ-2023-Function-Q16, PYQ-2023-Function-Q19, PYQ-2023-Function-Q17

$\quad$ If function is in the form:

• $\sqrt{\mathrm{f}(\mathrm{x})}$, $\quad$ take $\mathrm{f}(\mathrm{x}) \geq 0$

• $\frac{1}{\sqrt{\mathrm{f}(\mathrm{x})}}$, $\quad$ take $\mathrm{f}(\mathrm{x})>0$

• $\frac{1}{\mathrm{f}(\mathrm{x})}$, $\quad$ take $\mathrm{f}(\mathrm{x}) \neq 0$

Properties Of Even and Odd Function:

• The graph of an even function is always symmetric about y-axis.

• The graph of an odd function is always symmetric about origin.

• Product of two even or odd function is an even function.

• Sum & difference of two even (odd) function is an even (odd) function.

• Product of an even or odd function is an odd function.

• Sum of even and odd function is neither even nor odd function.

• Zero function, i.e. f(x) = 0, is the only function which is both even and odd.

• If f(x) is an odd (even) function, then f’(x) is even (odd) function provided f(x) is differentiable on R.

• A given function can be expressed as sum of even and odd function.

$\quad$ $\quad$ i.e. $f(x)=\frac{1}{2}[f(x)+f(-x)]+\frac{1}{2}[f(x)-f(-x)]=$ even function + odd function

Properties Of Periodic Function:

• If a function $f(x)$ has period T, then period of $f(x n+a)=T / n$ and period of $f(\frac{x}{n}+a)=n T$.

• If the period of $f(x)$ is $T_1 $ and $ T_2$ has $T_2$ then the period of $f(x) \pm g(x)$ will be L.C.M. of $T_1 $ and $ T_2$ provided it satisfies definition of periodic function.

• If period of $f(x) $ and $ g(x)$ are same T, then the period of $af(x)+bg(x)$ will also be $T$.

Properties of Greatest Integer Function

PYQ-2023-Function-Q1, PYQ-2023-Function-Q17, PYQ-2023-Function-Q18, PYQ-2023-Definite_Integration-Q9, PYQ-2023-Definite_Integration-Q13, PYQ-2023-Limits-Q2

• $ [x+n]=n+[x], n \in I$

• $ [-x]=-[x], x \in I$

• $ [-x]=-[x]-1, x \notin I$

• $ [x] \geq n \Rightarrow x \geq n, n \in I$

• $ [x]>n \Rightarrow x \geq n+1, n \in I$

• $ [x] \leq n \Rightarrow x<n+1, n \in I$

• $ [x]<n \Rightarrow x<n, n \in I$

Properties of Fractional Part Function

PYQ-2023-Function-Q1

• $ {x}=x-[x]$

• $ {x}=x$, if $0 \leq x<1$

• $ {x}=0$, if $x \in I$

• $ {-x}=1-{x}$, if $x \notin I$

Strictly Increasing Function:

$\quad$ If $f^{\prime}>0$ for all $x $ in the interval, then the function $f$ is strictly increasing.

Strictly Decreasing Function:

$\quad$ If $f^{\prime}<0$ for all $x$ in the interval, then the function $f$ is strictly decreasing.

Constant Function:

$\quad$ If $f^{\prime}=0$ for all $x$ in the interval, then the function $f$ is constant.

Logarithmic Function:

PYQ-2023-Definite_Integration-Q8, PYQ-2023-Trigonometric_Ratios-Q4, PYQ-2023-Function-Q5, PYQ-2023-Function-Q9

$\quad$ A logarithmic function may be given by $y=f(x)=log_a x$ , where $a > 0, a \neq 1, x > 0$

• The graph of the function is increasing, if $a > 1$ and decreasing, if $0<a<1$

Exponential Function:

PYQ-2023-Definite_Integration-Q5, PYQ-2023-Function-Q9

$\quad$ Exponential function is given by $y=f(x)=a^x$ , where $a>0, a\neq 1$

• The graph of the exponential function is increasing,if $a>1$ and decreasing, if $0<a<1$

Domain Of Composite Function:

PYQ-2023-Function-Q4, PYQ-2023-Function-Q14, PYQ-2023-Hyperbola-Q2, PYQ-2023-Limits-Q4

$\quad$ The domain of a composite function $f(g(x))$ is the set of those inputs $x$ in the domain of $g$ for which $g(x)$ is in the domain of $f$.