Functions-and-Types-of-Functions
Functions and Types of Functions
1. What Are Functions in Mathematics?
Definition: A function is a relation between a set of inputs (domain) and a set of permissible outputs (co-domain) such that each input is associated with exactly one output.
Sets A and B: For a relation from set A to set B to qualify as a function, every element in set A must map to one and only one element in set B.
Characteristics:
Each element of set A is associated with only one element of set B.
No two pairs can have the same first element.
2. Conditions for a Function
Both sets A and B must be non-empty.
Domain: Set A (input values)
Co-domain: Set B (possible output values)
Unique Mapping: For each element "a" in set A, there is a unique element "b" in set B such that f(a) = b.
Example: For a function f(x) = x^2, if the input x is 2, then the output is 4: f(2) = 4.
Range: The range is the set of all outputs from the function, a subset of set B.
3. Types of Functions
Real-Valued Function: A function where both inputs and outputs are real numbers.
Examples of Functions:
f(x) = 2: Inputs are {1, 2, 3} with outputs {2, 4, 6}, Range = {2, 4, 6}.
g(x) = x^2: Inputs {-2, -1, 0, 1, 2} with outputs {4, 1, 0, 1, 4}, Range = {0, 1, 4}.
4. Vertical Line Test
Used to determine if a curve is a function.
If a vertical line intersects the curve more than once, then it is not a function.
5. Representation of Functions
Functions can be represented as f(x).
Example: f(x) = x^3 indicates f of x is equal to x cubed.
6. Steps for Solving Functions
Example 1: Find output for g(t) = 6t^2 + 5:
At t = 0: g(0) = 5
At t = 2: g(2) = 29
Example 2: For f(t) = 3t + 7:
At t = 1: f(1) = 10
At t = -3: f(-3) = -2
7. Types of Functions (Detailed)
One-one Function (Injective Function):
Definition: Each element in the domain has a distinct image in the co-domain.
Example: f(x) = 2x + 3.
Many-one Function:
Definition: At least two elements in the domain map to the same element in the co-domain.
Example: f(x) = x^2.
Case: f(2) = 4 and f(-2) = 4.
Onto Function (Surjective Function):
Definition: Every element in the co-domain is an image of at least one element in the domain.
Into Function:
Definition: Not every element in the co-domain has a corresponding element in the domain.
8. Polynomial Functions
Definition: A mathematical expression involving constants, variables, and exponents.
General form: f(x) = h_n*x^n + h_(n-1)*x^(n-1) + ... + h_0, where n is non-negative and h coefficients are constants.
Types of Polynomial Functions Based on Degree:
Constant Function: Degree = 0. Example: f(x) = 5.
Linear Function: Degree = 1. Example: f(x) = mx + c.
Quadratic Function: Degree = 2. Example: f(x) = ax^2 + bx + c.
Cubic Function: Degree = 3. Example: f(x) = ax^3 + bx^2 + cx + d.
9. Identical Functions
Definition: Two functions f and g are identical if:
Same Domain: Df = Dg
Same Range: Rf = Rg
Same Values: f(x) = g(x) for all x in their domain.
10. Inverse Functions
Definition: A function that reverts the action of another function. If f(x) produces y, putting y into the inverse of f yields x.
Steps to find inverse:
Write y = f(x).
Solve for x in terms of y.
Switch x's and y's.
Result is y = f^(-1)(x).
11. Asymptotes
Definition: A line that a graph approaches but never touches.
Types of Asymptotes:
Horizontal Asymptotes: Describe end behavior.
Vertical Asymptotes: Occurs when the function approaches infinity at a specific x-value.
Oblique Asymptotes: Occurs when the function approaches a linear equation.
12. Determining Asymptotes for Rational Functions
Vertical asymptotes found by setting the denominator to zero.
Assess degrees of polynomials in the numerator and denominator to find horizontal asymptotes.