The range or rank of a function is the set of values that the function can assume. In other words, it is the set of y values that you get when you put all possible x values into the function. This set of possible values of x is called the domain. If you want to know how to find the rank of a function, just follow these steps.
Steps
Method 1 of 4: Finding the Rank of a Function Having a Formula
Step 1. Write the formula
Suppose it is the following: f (x) = 3 x2+ 6 x - 2. This means that, by inserting any x in the equation, the corresponding y value will be obtained. This is the function of a parable.
Step 2. Find the vertex of the function if it is quadratic
If you are working with a straight line or with a polynomial of an odd degree, for example f (x) = 6 x3 + 2 x + 7, you can skip this step. But, if you are working with a parabola or any equation where the x coordinate is squared or raised to an even power, you need to plot the vertex. To do this, just use the formula -b / 2a to get the x coordinate of the vertex of the function 3 x2 + 6 x - 2, where 3 = a, 6 = b and - 2 = c. In this case - b is -6 and 2 a is 6, so the x coordinate is -6/6 or -1.
- Now enter -1 in the function to get the y coordinate. f (-1) = 3 (-1)2 + 6(-1) - 2 = 3 - 6 - 2 = - 5.
- The vertex is (-1, - 5). Make the graph by drawing a point where the x coordinate is -1 and y is - 5. It should be in the third quadrant of the graph.
Step 3. Find some other points in the function
To get an idea of the function, you should substitute other x coordinates in order to get an idea of how the function looks, before even starting to search for the range. Since it is a parabola and the coefficient in front of the x2 is positive (+3), it will be facing up. But, just to give you an idea, let's insert some x coordinates in the function to see what y values it returns:
- f (- 2) = 3 (- 2)2 + 6 (- 2) - 2 = -2. A point on the graph is (-2; -2)
- f (0) = 3 (0)2 + 6 (0) - 2 = -2. Another point on the graph is (0; -2)
- f (1) = 3 (1)2 + 6 (1) - 2 = 7. A third point on the graph is (1; 7)
Step 4. Find the range on the graph
Now look at the y coordinates on the graph and find the lowest point where the graph touches a y coordinate. In this case, the lowest y coordinate is in the vertex, -5, and the graph extends to infinity above this point. This means that the range of the function is y = all real numbers ≥ -5.
Method 2 of 4: Find the Range on the Graph of a Function
Step 1. Find the minimum of the function
Find the minimum y coordinate of the function. Suppose the function reaches its lowest point at -3. y = -3 could also be a horizontal asymptote: the function could approach -3 without ever touching it.
Step 2. Find the maximum of the function
Suppose the function reaches its highest point at 10. y = 10 could also be a horizontal asymptote: the function could approach 10 without ever touching it.
Step 3. Find the rank
This means that the range of the function - the range of all possible y coordinates - ranges from -3 to 10. Thus, -3 ≤ f (x) ≤ 10. Here is the rank of the function.
- Suppose the graph reaches its lowest point at y = -3, but always goes up. Then the rank is f (x) ≥ -3.
- Suppose the graph reaches its highest point at 10, but always goes down. Then the rank is f (x) ≤ 10.
Method 3 of 4: Finding the Rank of a Relationship
Step 1. Write the report
A relationship is a set of ordered pairs of x and y coordinates. You can look at a relationship and determine its domain and range. Suppose you have the following relation: {(2, -3), (4, 6), (3, -1), (6, 6), (2, 3)}.
Step 2. List the y coordinates of the relationship
To find the rank, you simply have to write down all the y coordinates of each ordered pair: {-3, 6, -1, 6, 3}.
Step 3. Remove duplicate coordinates so that you only have one of each y coordinate
You will notice that you have listed "6" twice. Remove it, so that you are left with {-3, -1, 6, 3}.
Step 4. Write the rank of the relationship in ascending order
Now rearrange the numbers as a whole from smallest to largest, and you will have the rank of the relation {(2; -3), (4; 6), (3; -1), (6; 6), (2; 3)}: {-3; -1; 3; 6}. That's all.
Step 5. Make sure the relationship is a function
For a relation to be a function, every time you have a certain x coordinate you must have the same y coordinate. For example, the relation {(2, 3) (2, 4) (6, 9)} is not a function, because when you put 2 as x, the first time you get 3, while the second time you get 4. For a relation to be a function, if you enter the same input, you should always get the same result in output. If, for example, you enter -7, you should get the same y coordinate every time, whatever that is.
Method 4 of 4: Finding the Rank of a Function Spelled Out by a Problem
Step 1. Read the problem
Suppose you are working with the following problem: Barbara is selling tickets to her school play for 5 euros each. The amount of money you collect is a function of how many tickets you sell. What is the range of the function?
Step 2. Write the problem in the form of a function
In this case, M represents the amount of money Barbara collects and t the amount of tickets she sells. Since each ticket costs 5 euros, you will need to multiply the amount of tickets sold by 5 to find the amount of money. Therefore the function can be written as M (t) = 5 t.
For example, if Barbara sells 2 tickets, you have to multiply 2 by 5 to get 10, the amount of euros you get
Step 3. Determine the domain
To determine the rank, you must first find the domain. The domain consists of all possible values of t that can be inserted into the equation. In this case, Barbara can sell 0 tickets or more - she cannot sell negative tickets. Since we do not know the number of seats in your school's auditorium, we can assume that you can theoretically sell an infinite number of tickets. And he can only sell full tickets: he cannot sell half a ticket, for example. Therefore the domain of the function is t = any non-negative integer.
Step 4. Determine the rank
The codomain is the possible amount of money Barbara can get from her sale. You have to work with the domain to find the rank. If you know that the domain is any non-negative integer and that the formula is M (t) = 5t, then you know that it is possible to insert any non-negative integer into this function to get the set of outputs or rank. For example, if he sells 5 tickets, then M (5) = 5 x 5 = 25 euros. If you sell 100, then M (100) = 5 x 100 = 500 euros. Consequently, the rank of the function is any non-negative integer that is a multiple of 5.
This means that any non-negative integer that is a multiple of five is a possible output for the function's input
Advice
- See if you can find the inverse of the function. The domain of the inverse of a function is equal to the rank of that function.
- Check to see if the function repeats. Any function that repeats along the x axis will have the same rank for the entire function. For example, f (x) = sin (x) has a rank between -1 and 1.
