Module 3 Reading Material on Functions PDF

Week 3 Reading Material ======================= Definition: ----------- For any sets S and T we say that a function f from (or "mapping") S to T (f: S -\> T) is a particular assignment of exactly one element of T to each element of S. Function Representation (Block Diagram): -------------------...

Week 3 Reading Material ======================= Definition: ----------- For any sets S and T we say that a function f from (or "mapping") S to T (f: S -\> T) is a particular assignment of exactly one element of T to each element of S. Function Representation (Block Diagram): ----------------------------------------- [Example]: set of students S = {s1, s2, s3, s4, s5, s6, s7, s8} set of sections T = {A, B, C, D, E, NC} and let f be a function that maps the students to their respective sections defined by\ f: S -\> T. Will all sorts of assignments/ mappings form functions in the above example? What are necessary conditions for a mapping to be a function? ------------------------------------------------------------- For a mapping to be called as a function, there are two necessary conditions: 1. Every element from the domain must have an image (or) No element in the domain must be left out. 2. No element in the domain can have more than one image. [Example]: In the second instance, the element b has two images. In the third instance, the element d does not have an image. Which do not obey the necessary conditions for functions, hence their corresponding mappings do not form functions. Whereas, in the first instance, both the conditions are met and therefore, we can call the mapping as a function. Terminology Related to Functions: --------------------------------- 1. Domain[\ ] The domain of a function is defined as the set of values that one can serve as an input to the function in order to get a finite output is called as a domain. 2. Range: ------- The range can be defined as the set of all outputs of a function that we get after substituting the values from the domain. How to find Domain function? ---------------------------- - The domain of any polynomial function such as a linear function, quadratic function, cubic function, etc. is a set of all real numbers (R).  - The domain of a logarithmic function f(x) = log x is x \> 0 or (0, ∞).  - The domain of a square root function f(x) = √x is the set of non-negative real numbers which is represented as \[0, ∞). - The domain of an exponential function is the set of all real numbers (R). - A rational function is defined only for non-zero values of its denominator. So, to determine the domain of a rational function y = f(x), set the denominator ≠ 0. - Domain of the Polynomial functions (linear, quadratic, cubic, etc) function is R (all real numbers). - Domain of the square root function √x is x ≥ 0. - Domain of the exponential function is R. - Domain of the logarithmic function is x \> 0. - We know that, the domain of a rational function y = f(x), denominator ≠ 0. 1. Consider the function f(x) = [\$\\frac{4}{x\\ --\\ 2}\\ \$]{.math.inline} find the domain of f (x)? 2. Find the domain of the function f (x) = ln (x^2^ + x -- 2) If we consider f (x) = ln (x^2^ + x -- 2), then the logarithmic function is defined only when x^2^ + x -- 2 \> 0 - We can express this x^2^ + x -- 2 as x^2^ + 2x -- x -- 2 \> 0 - x (x + 2) -- 1 (x + 2) \> 0 - (x -- 1) (x + 2) \> 0 - (x -- 1) (x -- (- 2)) \> 0 - This is of the form of (x -- a) (x -- b) \> 0 such that a \< b - Then the domain is given by (- ∞, a) u (b, ∞) Interval Notation and Descriptions: ----------------------------------- +-----------------+-----------------+-----------------+-----------------+ | **inequality** | **Interval | **Graph on | **Description** | | | notation** | number line** | | +=================+=================+=================+=================+ | *x* \> *a* | (a, ∞) | ( | *x* is greater | | | | | than *a* | | | | a | | +-----------------+-----------------+-----------------+-----------------+ | *x* \< *a* | (-∞,a) | ) | *x* is less | | | | | than *a* | | | | a | | +-----------------+-----------------+-----------------+-----------------+ | *x* ≥ *a* | \[a, ∞) | \[ | *x* is greater | | | | | than or equal | | | | a | to *a* | +-----------------+-----------------+-----------------+-----------------+ | *x* *≤ a* | (-∞,a\] | \] | *x* is less | | | | | than or equal | | | | a | to *a* | +-----------------+-----------------+-----------------+-----------------+ | a \< x \< b | (a,b) | ( ) | x is strictly | | | | | between a and b | | | | a b | | +-----------------+-----------------+-----------------+-----------------+ | a *≤ x \< b* | \[a,b) | \[ ) | x is between a | | | | | and b, to | | | | a b | include a | +-----------------+-----------------+-----------------+-----------------+ | a *\< x ≤ b* | (a,b\] | ( \] | x is between a | | | | | and b, to | | | | a b | include b | +-----------------+-----------------+-----------------+-----------------+ | a *≤* *x ≤ b* | \[a,b\] | \[ \] | x is between a | | | | | and b, to | | | | a b | include a and b | +-----------------+-----------------+-----------------+-----------------+ How to find the Range of a function? ------------------------------------ To determine the range of a function f(x), we equate the function to an arbitrary variable y as follows: 1. **Express the function in terms of y:** Start with the relationship y = f(x). 2. **Find the inverse function:** Rewrite the equation in terms of x by solving f(x) for x in terms of y. 3. **Evaluate the domain of the inverse:** Determine the set of all possible values y can take for the inverse function x = f^−1^(y). 4. **Relate to the original function:** The domain of the inverse function corresponds to the range of the original function. [Example]: 1. Consider the function f(x) = [\$\\frac{4}{x\\ --\\ 2}\\ \$]{.math.inline} find the Range of f (x)? - y = [\$\\frac{4}{x\\ --\\ 2}\$]{.math.inline} - y (x -- 2) = 4 - yx -- 2y = 4 - yx = 4 + 2y - x = [\$\\frac{(4 + 2y)}{y}\$]{.math.inline} - The domain of the function [\$\\frac{(4 + 2y)}{y}\$]{.math.inline} is R -- {0} Since, for all values of y, the output of the function is finite except of y = 0 3. Codomain: The set of all "**possible**" outputs of a function is called as the codomain. Representation: --------------- Differences between the Range and Codomain of a function: --------------------------------------------------------- **Aspect** **Domain** **Range** ---------------------- ------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------- **Definition** The set of all possible output values of a function, regardless of whether they are actually attained by the function. The set of all actual output values that the function produces from the given inputs. **Determined By** Defined by the function's definition Determined by the specific values in the domain and the function's behaviour **Containment** Includes all elements that could potentially be outputs. Only includes elements that are actual outputs. **Example** For a function *f(x)* = *x^2^,* if the codomain is ℝ, it includes all real numbers. For the same function *f(x)* = *x^2^*, if the domain is ℝ, the range is \[0, ∞), only including non- negative real numbers. **Role in Function** Sets the scope of potential outputs. Describes the set of outputs produced by the function. **Key Points:** - **Codomain:** The codomain is defined as part of the function\'s definition. It sets the \'target\' space in which outputs lie but does not specify which elements of this set are actually outputs. - **Range:** The range, however, is the set of outputs that are actually attained by the function when applied to the elements of its domain 4. Image: If we consider a function such that f(x) = y, then y is called as the image of x. 5. Pre-Image: ----------- If we consider a function such that f(x) = y, then x is called as the pre-image of y. [Example]: If we consider the function f (x) = √x, and consider the value for x as 25 from the domain, Then f (25) = √25 = 5 Here, 25 is the Pre-image of 5 Types of Functions: ------------------- Floor function: --------------- The floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). ------------------------------------------------------------------------------------------------------------------------------------------------------------------ Example: ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3. 2. Ceiling Function: ----------------- the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). Example: ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2. 3. Injective function (one-to-one function): ----------------------------------------- It is defined as a function f: S -\> T such that the values of f(a) = f(b) if and only if a = b for all the values of a and b from the domain S. Injective functions are also called as one-to-one functions or simply injection. 4. Surjective function: -------------------- A function f: S -\> T is surjective if and only if for every element from T, there is a pre-image in S. A surjective function is also called as an onto function or simply surjection. [Note: ] 1. The range of a surjective function is equal to its codomain. 2. An onto function maps the set S onto the entirety of T not just a part of it. 5. Bijective function: ------------------- A function f: S -\> T is a bijective function if and only if it is both an injection and a surjection. How to check for Injectivity, Surjectivity, & Bijectivity of a function? ------------------------------------------------------------------------ How to check for Injectivity of a function: ------------------------------------------- We consider f (a) = f (b) for the function and evaluate the equality. If we end up with a situation a = b, then the function is injective, else it is not injective. [Example]: 1. Consider the function f: R -\> R defined as f (x) = x^3^. Is this function injective? Let us consider, f (a) = (a)^3^ f (b) = b^3^ Let us assume that f (a) = f (b) - a^3^ = b^3^ - a = b Therefore, we can conclude that f (x) = x^3^ is Injective. 2. Consider the function f: R -\> R defined as f (x) = x^2^. Is this function Injective? Let us consider, f (a) = (a)^2^ f (b) = (b)^2^ Let us assume that f (a) = f (b) - a^2^ = b^2^ - Here, we have various possibilities for the values that a and b can take - The possibilities are: - a is negative, b is negative =\> - a = - b - a is positive, b is positive. =\> a = b - a is positive, b is negative. =\> a = - b - a is negative, b is positive. =\> - a = b Analogy for understanding this scenario: ---------------------------------------- How to check for Surjectivity of a function: -------------------------------------------- A function is said to be a surjective function if the Co-Domain = Range of the function. [Example]: 1. Consider the function f (x) = x^2^ such that f: R -\> R. Is f a surjective function or not? - The co-domain of the function as highlighted is R (set of real numbers) - y = x^2^ - x = √y - The domain of the inverse function is R^+^ ∪ {0} - The range of f (x) = R^+^ ∪ {0} Real- Valued functions: ------------------------ A function that has either R or one of its subsets as its range is called as a real- valued function. Identity function: ------------------ Let A be a set, then the identity function on a is defined as follows: I~A~ : A -\> A, where, I~A~(x) = x for all values of x belongs to A. Note: an identity function is both injective as well as surjective. Therefore, it is bijective. Function Operators: ------------------- Functions with overlapping domains can be added, subtracted, multiplied and divided.  If   f(x) and g(x) are two functions, then for all x in the domain of both functions the sum, difference, product and quotient are defined as follows: ***( f + g ) (x) = f(x) + g(x)*** ***( f - g ) (x) = f(x) - g(x)*** ***(fg) (x) = f(x) × g(x)*** [\$\\left( \\frac{\\mathbf{f}}{\\mathbf{g}} \\right)\$]{.math.inline}***(x) =*** [\$\\frac{\\mathbf{f}\\left( \\mathbf{x} \\right)}{\\mathbf{g}\\left( \\mathbf{x} \\right)}\$]{.math.inline}***, g(x)*** [**≠**]{.math.inline} ***0*** [Example]: Perform the following operations on f (x) = 2 x^2^ − 4 and g (x) = x^2^ + 4 x −2. a. (f + g) (x) = f (x) + g (x) - (f + g) (x) = 2 x^2^ -- 4 + x^2^ + 4 x −2 - (f + g) (x) = 3 x^2^ + 4 x -- 6. b. (f -- g) (x) = f (x) -- g (x) - (f -- g) (x) = (2 x^2^ -- 4) -- (x^2^ + 4 x −2) - (f -- g) (x) = x^2^ -- 4 x -- 2 c. (f.g) (x) = f(x). g(x) - (f.g) (x) = (2 x^2^ -- 4). (x^2^ + 4 x −2) - (f.g) (x) = 2 x^4^ + 8 x^3^ − 4x^2^ -- 16 x + 8. d. (f/g) (x) = [\$\\frac{f\\left( x \\right)}{g(x)}\$]{.math.inline} - (f/g) (x) = [\$\\frac{2\\ x2\\ --\\ 4}{\\left( x2\\ + \\ 4\\ x\\ - 2 \\right)}\$]{.math.inline} Composition of Functions: ------------------------- The process of combining functions such that the output of one function becomes the input of the other is called as composition of functions. The resulting function is called as a composite function. We represent it by the following notation: ***f**o**g(x) = f(g(x))*** [Example]: Given the functions f(x) = 3x -- 5 and g(x) = x^2^ + 2, evaluate for: a. (f o g) (2) b. (g o f) (-1) a. (f o g) (x) = f (g (x)) - f (x^2^ + 2) = 3 (x^2^ + 2) -- 5 - (f o g) (x) = 3 x^2^ + 6 -- 5 = 3 x^2^ + 1 - (f o g) (2) = 3 (2)^2^ + 1 = 3 (4) + 1 = 12 + 1 = 13. b. (g o f) (x) = g (f (x)) - (g o f) (x) = g (f (x)) = g (3x -- 5) = (3x -- 5)^2^ + 2 - (g o f) (x) = 9 x^2^ + 25 -- 30 x + 2 = 9 x^2^ -- 30 x + 27 - (g o f) (-1) = 9 (- 1)^2^ -- 30 (- 1) + 27 = 9 + 30 + 27 = 66. Inverse of a function: ---------------------- Let f: S -\> T be a bijective function, then the inverse function is a function obtained by reversing the function f. Here, g: T -\> S is the inverse of the function f. The inverse of a function is denoted by f ^-1^ if f is defined as f(x) = y, then f ^-1^ is defined as F ^--\ 1^(y) = x [Note:] F ^-1^o f: S -\> S That is, (f^-1^of) = I~S~ that means the composition operation upon the function and the inverse of it would result in an identity function. [Example]: Consider the function f (x) = 2x + 3. Find the inverse of the function. [Sol]: Let, y = f (x) - y = 2 x + 3 - 2 x = y -- 3 - x = [\$\\frac{(y\\ --\\ 3)}{2}\\ \$]{.math.inline} Here, x = f^-1^(y) is the inverse function of f (x).