## Relations & Functions

# Functions

- A relation f is defined from A to B is said to be a function if every element of set A is associated with unique element of set B.
- Throughout this topic function means real valued function.
- The graph of the curve is said to be a function if the vertical line cuts the graph at exactly one point.
- The number of functions that can be defined from the set A to the set B is n(B)
^{n(A)}. - Range is the subset of co-domain
- The projection of the graph y = f(x) on the x-axis is equal to the domain of ‘f’ whereas the projection on the y-axis is equal to the range of ‘f’
- (x − a)(x − b) > 0 ⇒ x < a or x > b for a < b
- (x − a)(x − b) < 0 ⇒ a < x < b for a < b
- Domain of a function f: A → B:

→ If f: A → B is a function then A is called domain

→ The set of all values of x at which the function y = f(x) is defined is called domain

→ The graph covering values on x-axis is called domain. - Range of the function f: A → B

→ If f: A → B is a function then f(A) is called Range.

→ The set of all values of y at which the function y = f(x) is defined is called Range.

→ The graph covering values on y-axis is called Range. - Codomain of the function f: A → B:

→ If f: A → B is a function then B is called codomain of ‘f’. - If the domain of f = A and the domain of g = B, then

→ Domain of f ± g is A ∩ B

→ Domain of fg is A ∩ B

→ Domain of \frac{f}{g} is A ∩ B and g ≠ 0 - Domain of \sqrt{f(x)} = {x / f(x) ≥ 0}
- Domain of \frac{1}{\sqrt{f(x)}} = {x / f(x) > 0}
- Domain of \frac{1}{f(x)} = {x / f(x) ≠ 0}
- Domain of \sqrt{|f(x)|}=R
- Domain of \frac{1}{\sqrt{|f(x)|}} = {x / f(x) ≠ 0}
- Domain of \log_{g(x)}^{f(x)} = {x / f(x) > 0, g(x) > 0 and g(x) ≠ 1}
- Domain of \frac{1}{\log|f(x)|} = {x / f(x) ≠ –1, 0, 1}

- Even function: If f(−x) = f(x)
- Odd function: If f(−x) = −f(x).

**Short cuts on Domains:**

- The domain of \sqrt{a^{2}-x^{2}} is [− a, a]
- The domain of \frac{1}{\sqrt{a^{2}-x^{2}}} is (− a, a)
- The domain of \sqrt{x^{2}-a^{2}} is (−∞,−a] ∪ [a, ∞)
- The domain of \frac{1}{\sqrt{x^{2}-a^{2}}} is (−∞,−a) ∪ (a, ∞)
- The domain of \sqrt{(x-a)(x-b)} when a < b is (−∞,−a] ∪ [b, ∞)
- The domain of \frac{1}{\sqrt{(x-a)(x-b)}} when a < b is (−∞,−a) ∪ (b, ∞)
- The domain of \sqrt{(x-a)(b-x)} when a < b is [a, b]
- The domain of \frac{1}{\sqrt{(x-a)(b-x)}} when a < b is (a, b)
- The domain of \sqrt{\frac{x-a}{x-b}} when a < b is (−∞, a] ∪ (b, ∞)
- The domain of \sqrt{\frac{x-a}{x-b}} when a > b is (−∞, b) ∪ [a, ∞)
- The domain of \sqrt{\frac{x-a}{b-x}} when a < b is [a, b)
- The domain of \sqrt{\frac{x-a}{b-x}} when a > b is (b, a]
- The domain of log (a
^{2}– x^{2}) is (−a, a) - The domain of log (x
^{2}– a^{2}) is (−∞,−a) ∪ (a, ∞) - The domain of f(x)=\frac{|x+c|}{x+c} is R − {−C}

**Short cuts on Ranges:**

- Range of f(x)=\sqrt{a^{2}-x^{2}} is [0, a]
- The range of the function f(x) = a cos x + b sin x + c is \left[c-\sqrt{a^{2}+b^{2}},c+\sqrt{a^{2}+b^{2}}\right]
- The range of the function f(x) = \frac{|x+c|}{x+c} is {−1, 1}
- The range of the function f(x) = ax
^{2}+ bx + c when a < 0 is \left(-\infty, \frac{4ac-b^{2}}{4a}\right] - The range of the function f(x) = ax
^{2}+ bx + c when a > 0 is \left[\frac{4ac-b^{2}}{4a},\infty\right) - The range of the function f(x)=\sqrt{x-a}+\sqrt{b-x} ∀ x ∈ [a, b] is \left[\sqrt{b-a},\sqrt{2(b-a)}\right]
- The range of the function f(x)=\sqrt{x-a}-\sqrt{b-x} ∀ x ∈ [a, b] is \left[-\sqrt{b-a},\sqrt{b-a}\right]
- The range of ax+\frac{b}{x} is \left(-\infty,-2\sqrt{ab}\right]\cup\left[2\sqrt{ab},\infty\right)
- The range of a cot x + b tan x is \left[2\sqrt{ab},\infty\right)
- The range of a
^{2}sec^{2}x + b^{2}cosec^{2}x is \left[(a+b)^2,\infty\right) - The range of f(x) = a sin
^{2}x + b cosec^{2}x is \left[2\sqrt{ab},\infty\right) - The range of f(x) = a tan
^{2}x + b cot^{2}x is \left[2\sqrt{ab},\infty\right) - The range of f(x) = a sec
^{2}x + b cos^{2}x is \left[2\sqrt{ab},\infty\right) - The range of f(x) = a sin
^{2}x + b sin x cos x + c cos^{2}x is \left[\frac{a+c}{2}-\frac{\sqrt{b^{2}+(a-c)^2}}{2},\frac{a+c}{2}+\frac{\sqrt{b^{2}+(a-c)^2}}{2}\right] - The range of f(x) = a sinx + b cosec x is \left[2\sqrt{ab},\infty\right)
- The range of f(x) = a cos x + b sec x is \left[2\sqrt{ab},\infty\right)
- Two functions f and g are said to be identical (or) equal functions if

(i) Domain of f = Domain of g

(ii) Codomain of f = codomain of g

(iii) Range of f = Range of g. - Modulus function (or) Absolute value function: The function f : R → R defined as

f(x)=|x|=\begin{cases}x, & {\tt if} \ x > 0 \\0 & {\tt if} \ x = 0\\-x & {\tt if} \ x < 0\end{cases} is called Absolute value function.

Domain = R and Range = [0, ∞) - \tt \mid x\pm y\mid\ \leq\ \mid x\mid + \mid y \mid

\tt \mid x\pm y\mid\ \geq\ \bigm\vert \mid x\mid -\mid y \mid \bigm\vert

\tt \bigm\vert \mid x\mid -\mid y \mid \bigm\vert\ \leq\ \mid x \pm y\mid\ \leq\ \mid x\mid + \mid y \mid - Greatest integer (Step (or) Integral) function: The function f : R → R defined as f(x) = [x] is called greatest integer function, where [x]=\begin{cases}k & k \leq x < k + 1\\k + 1 & k+1 \leq x < k+2\end{cases} where k ∈ z

Domain = R and Range = Z - [x] ≤ x < [x] + 1
- [x]+[-x]=\begin{cases}0 & {\tt if} \ \ x \in I\\-1 & {\tt if} \ \ x \notin I \end{cases}
- [n + x] = n + [x] where n is integer
- n ≤ x < n + 1 ⇔ [x] = n where n is integer
- Fractional part function: A function f : R → R defined as f(x) = {x} where {x} = x − [x] is called fractional part function

Domain = R and Range = [0, 1) - [{x}] = 0
- \left\{x\right\}+\left\{-x\right\}=\begin{cases}0 \ ;& x \in Integer\\1 \ ; & x \notin Integer\end{cases}
- Signum function: A function f : R → R defined as f(x) = \begin{cases}\frac{|x|}{x} \ ,& x \neq 0\\0 \ , & x = 0 \end{cases} is called signum function.

Domain = R and Range = {−1, 0, 1} - If f(x + y) = f(x) + f(y) ∀ x, y ∈ R, then f(x) = kx.
- If f(x + y) = f(x) f(y) ∀ x, y ∈ R, then f(x) = a
^{kx} - If f(x + y) = f(x) = f(y) ∀ x, y ∈ R, then f(x) = k.
- f(xy) = f(x) + f(y) ∀ x, y ∈ R, then f(x) = k log
_{e}x. - f(xy) = f(x)f(y) ∀ x, y ∈ R, then f(x) = x
^{n}. - If f(x) . f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right), then f(x) = 1 ± x
^{n}. **Tricks for problem solving:**- The total number of functions from set A to set B containing ‘m’ and ‘n’ elements respectively is n
^{m}. - If n(x) = m; n(y) = n then one-one functions n
_{pm}if n ≥ m in case of n < m injections are ‘O’. - The graph of an odd function is symmetric about origin and it is placed either in the first and third quadrant or in the second and fourth quadrant.
- The graph of an even function is symmetric about y-axis.
- If f(x) is an odd function, then f’(x) is even function provided f(x) is differentiable on R.

### View the Topic in this video From 15:56 To 51:20

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**Algebra of functions** For functions f : X → **R** and g : X → **R**, we have

(f + g) (x) = f(x) + g(x), x ∈ X

(f − g) (x) = f (x) − g (x), x ∈ X

(f.g) (x) = f (x) .g (x), x ∈ X

(kf) (x) = k f (x)), x ∈ X, where k is a real number.

\tt \left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)},x \in X, g\left(x\right) \neq 0