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 (a² – x²) is (−a, a)
• The domain of log (x² – a²) 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) = ax² + bx + c when a < 0 is \left(-\infty, \frac{4ac-b^{2}}{4a}\right]
• The range of the function f(x) = ax² + 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 a² sec²x + b² cosec²x is \left[(a+b)^2,\infty\right)
• The range of f(x) = a sin²x + b cosec²x is \left[2\sqrt{ab},\infty\right)
• The range of f(x) = a tan²x + b cot²x is \left[2\sqrt{ab},\infty\right)
• The range of f(x) = a sec²x + b cos²x is \left[2\sqrt{ab},\infty\right)
• The range of f(x) = a sin²x + b sin x cos x + c cos²x 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_ex.
• f(xy) = f(x)f(y) ∀ x, y ∈ R, then f(x) = xⁿ.
• If f(x) . f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right), then f(x) = 1 ± xⁿ.
• 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.