Cartesian products of sets
Definition : Given two non-empty sets A and B, the set of all ordered pairs (x, y), where x ∈ A and y ∈ B is called Cartesian product of A and B; symbolically, we write A × B = {(x, y) | x ∈ A and y ∈ B} If A = {1, 2, 3} and B = {4, 5}, then A × B = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)} and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} (i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal, i.e. (x, y) = (u, v) if and only if x = u, y = v. (ii) If n(A) = p and n (B) = q, then n (A × B) = p × q. (iii) A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.
Relations
A Relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R. For example, the set R = {(1, 2), (– 2, 3), ( 1 2 , 3)} is a relation; the domain of R = {1, – 2, 1 2 } and the range of R = {2, 3}.
Functions
A relation f from a set A to a set B is said to be function if every element of set A has one and only one image in set B.
In other words, a function f is a relation such that no two pairs in the relation has the same first element.
The notation f : X →Y means that f is a function from X to Y. X is called the domain of f and Y is called the co-domain of f. Given an element x ∈ X, there is a unique element y in Y that is related to x. The unique element y to which f relates x is denoted by f (x) and is called f of x, or the value of f at x, or the image of x under f.
The set of all values of f(x) taken together is called the range of f or image of X under f.
Symbolically.
range of f = { y ∈ Y | y = f (x), for some x in X}
Definition : A function which has either R or one of its subsets as its range, is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.
Every function contains two types of variables: independent variables and dependent variables, whose values literally “depend” on the independent variables. For example, in the function y = f(x) = 2x + y, x is independent and y is dependent (in other words, y is a function of x). The valid values for a given independent variable x are collectively called the “domain.” The valid values for a given dependent variable y are collectively called the “range
1) Determine the type of function you’re working with.
The domain of the function is all of the x-values (horizontal axis) that will give you a valid y-value output. The function equation may be quadratic, a fraction, or contain roots. To calculate the domain of the function, you must first evaluate the terms within the equation.
Definition : Given two non-empty sets A and B, the set of all ordered pairs (x, y), where x ∈ A and y ∈ B is called Cartesian product of A and B; symbolically, we write A × B = {(x, y) | x ∈ A and y ∈ B} If A = {1, 2, 3} and B = {4, 5}, then A × B = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)} and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} (i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal, i.e. (x, y) = (u, v) if and only if x = u, y = v. (ii) If n(A) = p and n (B) = q, then n (A × B) = p × q. (iii) A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.
Relations
A Relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R. For example, the set R = {(1, 2), (– 2, 3), ( 1 2 , 3)} is a relation; the domain of R = {1, – 2, 1 2 } and the range of R = {2, 3}.
Functions
A relation f from a set A to a set B is said to be function if every element of set A has one and only one image in set B.
In other words, a function f is a relation such that no two pairs in the relation has the same first element.
The notation f : X →Y means that f is a function from X to Y. X is called the domain of f and Y is called the co-domain of f. Given an element x ∈ X, there is a unique element y in Y that is related to x. The unique element y to which f relates x is denoted by f (x) and is called f of x, or the value of f at x, or the image of x under f.
The set of all values of f(x) taken together is called the range of f or image of X under f.
Symbolically.
range of f = { y ∈ Y | y = f (x), for some x in X}
Definition : A function which has either R or one of its subsets as its range, is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.
Every function contains two types of variables: independent variables and dependent variables, whose values literally “depend” on the independent variables. For example, in the function y = f(x) = 2x + y, x is independent and y is dependent (in other words, y is a function of x). The valid values for a given independent variable x are collectively called the “domain.” The valid values for a given dependent variable y are collectively called the “range
P-1) Finding the Domain of a Function
1) Determine the type of function you’re working with.
The domain of the function is all of the x-values (horizontal axis) that will give you a valid y-value output. The function equation may be quadratic, a fraction, or contain roots. To calculate the domain of the function, you must first evaluate the terms within the equation.
- A quadratic function has the form ax2 + bx + c:[2] f(x) = 2x2 + 3x + 4
- Examples of functions with fractions include: f(x) = (1/x), f(x) = (x + 1)/(x - 1), etc.
- Functions with a root include: f(x) = √x, f(x) = √(x2 + 1), f(x) = √-x, etc.
2) Write the domain with proper notation. Writing the domain of a function involves the use of both brackets [,] and parentheses (,). You use a bracket when the number is included in the domain and use a parenthesis when the domain does not include the number. The letter U indicates a union that connects parts of a domain that may be separated by a gap.[3]
- For example, a domain of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10.
- Always use parentheses if you are a using the infinity symbol, ∞. This is because infinity is a concept and not a number.
3) Draw a graph of the quadratic equation. Quadratic equations make a parabolic graph that either points up or down. Given that the parabola will continue infinitely outward on the x-axis, the domain of most quadratic function is all real numbers. Stated another way, a quadratic equation encompasses all of the x-values on the number line, making its domain R (the symbol for all real numbers).[4]
- To get an idea of the function choose any x-value and plug it into the function. Solving the function with this x-value will output a y-value. These x- and y-values are a coordinate (x, y) of the graph of the function.
- Plot this coordinate and repeat the process with another x-value.
- Plotting a few values in this fashion should give you a general idea of shape of the quadratic function.
4) Set the denominator equal to zero, if it’s a fraction. When working with a fraction, you can never divide by zero. By setting the denominator equal to zero and solving for x, you can calculate the values that will be excluded in the function.[5]
- For example: Identify the domain of the function f(x) = (x + 1)/(x - 1).
- The denominator of this function is (x - 1).
- Set it equal to zero and solve for x: x – 1 = 0, x = 1.
- Write the domain: The domain of this function cannot include 1, but includes all real numbers except 1; therefore, the domain is (-∞, 1) U (1, ∞).
- (-∞, 1) U (1, ∞) can be read as the set of all real numbers excluding 1.The infinity symbol, ∞, represents all real numbers. In this case, all real numbers greater than 1 and less than one are included in the domain.
5) Set the terms inside the radical to be greater than or equal to zero, if there’s a root function. You cannot take the square root of a negative number; therefore, any x-value that leads to a negative number must be excluded from the domain of that function.[6]
- For example: Identify the domain of the function f(x) = √(x + 3).
- The terms within the radical are (x + 3).
- Set them greater than or equal to zero: (x + 3) ≥ 0.
- Solve for x: x ≥ -3.
- The domain of this function includes all real numbers greater than or equal to -3; therefore, the domain is [-3, ∞).
P-2) Finding the Range of a Quadratic Function
1) Confirm that you have a quadratic function. A quadratic function has the form ax2 + bx + c: f(x) = 2x2 + 3x + 4. The shape of a quadratic function on a graph is parabola pointing up or down. There are different methods to calculating the range of a function depending on the type you are working with.
the easiest way to identify the range of other functions, such as root and fraction functions, is to draw the graph of the function using a graphing calculator.
2) Find the x-value of the vertex of the function. The vertex of a quadratic function is the tip of the parabola. Remember, a quadratic equation is of the form ax2 + bx + c. To find the x-coordinate use the equation x = -b/2a. This equation is a derivative of the basic quadratic function which represents the equation with a zero slope (at the vertex of the graph, the slope of the function is zero).
- For example, find the range of 3x2 + 6x -2.
- Calculate x-coordinate of vertex: x = -b/2a = -6/(2*3) = -1
3) Calculate the y-value of the vertex of the function. Plug the x-coordinate into the function to calculate the corresponding y-value of the vertex. This y-value denotes the edge of your range for the function.
- Calculate y-coordinate: y = 3x2 + 6x – 2 = 3(-1)2 + 6(-1) -2 = -5.
- The vertex of this function is (-1, -5).
4) Determine the direction of the parabola by plugging in at least one more x-value. Choose any other x-value and plug it into the function to calculate the corresponding y-value. If the y-value is above the vertex, the parabola continues to +∞. If the y-value is below the vertex, the parabola continues to -∞.
- Use the x-value -2: y = 3x2 + 6x – 2 = y = 3(-2)2 + 6(-2) – 2 = 12 -12 -2 = -2.
- This yields the coordinate (-2, -2).
- This coordinate tells you that the parabola continues above the vertex (-1, -5); therefore, the range encompasses all y-values above -5.
- The range of this function is [-5, ∞)
5) Write the range with proper notation. Like the domain, the range is written with the same notation. Use a bracket when the number is included in the domain and use a parenthesis when the domain does not include the number. The letter U indicates a union that connects parts of a domain that may be separated by a gap.[9]
- For example, a range of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10.
- Always use parentheses if you are a using the infinity symbol, ∞
P-3) Finding the Range of a Function Graphically
1) Graph the function. Oftentimes, it is easiest to determine the range of a function by simply graphing it. Many root functions have a range of (-∞, 0] or [0, +∞) because the vertex of the sideways parabola is on the horizontal, x-axis. In this case, the function encompasses all of the positive y-values if the parabola goes up, or all of the negative y-values if the parabola goes down. Fraction functions will have asymptotes that define the range.[10]
- Some root functions will start above or below the x-axis. In this case, the range is determined by the point the root function starts. If the parabola starts at y = -4 and goes up, then the range is [-4, +∞).
- The easiest way to graph a function is to use a graphing program or a graphing calculator.
- If you do not have a graphing calculator, you can draw a rough sketch of a graph by plugging x-values into the function and getting the corresponding y-values. Plot these coordinates on the graph to get an idea of the shape of the graph.
2) Find the minimum of the function. Once you have graphed the function, you should be able to clearly see the lowest point of the graph. If there is no obvious minimum, know that some functions will continue on to -∞.
- A fraction function will include all points except those at the asymptote. They often have ranges such as (-∞, 6) U (6, ∞).
3) Determine the maximum of the function. Again, after graphing, you should be able to identify the maximum point of the function. Some functions will continue on to +∞ and therefore, will not have a maximum.
4) Write the range with proper notation. Like the domain, the range is written with the same notation. Use a bracket when the number is included in the domain and use a parenthesis when the domain does not include the number. The letter U indicates a union that connects parts of a domain that may be separated by a gap.[11]
- For example, a range of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10.
- Always use parentheses if you are a using the infinity symbol, ∞.
