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The range of a function is the set of numbers that the function can produce. In other words, it is the set of yvalues that you get when you plug all of the possible xvalues into the function. This set of possible xvalues is called the domain. If you want to know how to find the range of a function, just follow these steps.
Steps
Method 1
Method 1 of 4:Finding the Range of a Function Given a Formula

1Write down the formula. Let's say the formula you're working with is the following: f(x) = 3x^{2} + 6x 2. This means that when you place any x into the equation, you'll get your y value. This is the function of a parabola.

2Find the vertex of the function if it's quadratic. If you're working with a straight line or any function with a polynomial of an odd number, such as f(x) = 6x^{3}+2x + 7, you can skip this step. But if you're working with a parabola, or any equation where the xcoordinate is squared or raised to an even power, you'll need to plot the vertex. To do this, just use the formula b/2a to get the x coordinate of the function 3x^{2} + 6x 2, where 3 = a, 6 = b, and 2 = c. In this case b is 6, and 2a is 6, so the xcoordinate is 6/6, or 1.
 Now, plug 1 into the function to get the ycoordinate. f(1) = 3(1)^{2} + 6(1) 2 = 3  6 2 = 5.
 The vertex is (1,5). Graph it by drawing a point where the x coordinate is 1 and where the ycoordinate is 5. It should be in the third quadrant of the graph.
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3Find a few other points in the function. To get a sense of the function, you should plug in a few other xcoordinates so you can get a sense of what the function looks like before you start to look for the range. Since it's a parabola and the x^{2} coordinate is positive, it'll be pointing upward. But just to cover your bases, let's plug in some xcoordinates to see what y coordinates they yield:
 f(2) = 3(2)^{2} + 6(2) 2 = 2. One 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).

4Find the range on the graph. Now, look at the ycoordinates on the graph and find the lowest point at which the graph touches a ycoordinate. In this case, the lowest ycoordinate is at the vertex, 5, and the graph extends infinitely above this point. This means that the range of the function is y = all real numbers ≥ 5.Advertisement
Method 2
Method 2 of 4:Finding the Range of a Function on a Graph

1Find the minimum of the function. Look for the lowest ycoordinate of the function. Let's say the function reaches its lowest point at 3. This function could also get smaller and smaller infinitely, so that it doesn't have a set lowest point  just infinity.

2Find the maximum of the function. Let's say the highest ycoordinate that the function reaches is 10. This function could also get larger and larger infinitely, so it doesn't have a set highest point  just infinity.

3State the range. This means that the range of the function, or the range of ycoordinates, ranges from 3 to 10. So, 3 ≤ f(x) ≤ 10. That's the range of the function.
 But let's say the graph reaches its lowest point at y = 3, but goes upward forever. Then the range is f(x) ≥ 3 and that's it.
 Let's say the graph reaches its highest point at 10 but goes downward forever. Then the range is f(x) ≤ 10.
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Method 3
Method 3 of 4:Finding the Range of a Function of a Relation

1Write down the relation. A relation is a set of ordered pairs with of x and y coordinates. You can look at a relation and determine its domain and range. Let's say you're working with the following relation: {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}.^{[1] X Research source }

2List the ycoordinates of the relation. To find the range of the relation, simply write down all of the ycoordinates of each ordered pair: {3, 6, 1, 6, 3}.^{[2] X Research source }

3Remove any duplicate coordinates so that you only have one of each ycoordinate. You'll notice that you have listed "6" two times. Take it out so that you are left with {3, 1, 6, 3}.^{[3] X Research source }

4Write the range of the relation in ascending order. Now, reorder the numbers in the set so that you're moving from the smallest to the largest, and you have your range. The range of the relation {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)} is {3,1, 3, 6}. You're all done.^{[4] X Research source }

5Make sure that the relation is a function. For a relation to be a function, every time you put in one number of an x coordinate, the y coordinate has to be the same. For example, the relation {(2, 3) (2, 4) (6, 9)} is not a function, because when you put in 2 as an x the first time, you got a 3, but the second time you put in a 2, you got a four. For a relation to be a function, if you put in the same input, you should always get the same output. If you put in a 7, you should get the same y coordinate (whatever it may be) every single time.^{[5] X Research source }Advertisement
Method 4
Method 4 of 4:Finding the Range of a Function in a Word Problem

1Read the problem. Let's say you're working with the following problem: "Becky is selling tickets to her school's talent show for 5 dollars each. The amount of money she collects is a function of how many tickets she sells. What is the range of the function?"

2Write the problem as a function. In this case, M represents the amount of money she collects, and t represents the amount of tickets she sells. However, since each ticket will cost 5 dollars, you'll have to multiply the amount of tickets sold by 5 to find the amount of money. Therefore, the function can be written as M(t) = 5t.
 For example, if she sells 2 tickets, you'll have to multiply 2 by 5 to get 10, the amount of dollars she'll get.

3Determine the domain. To determine the range, you must first find the domain. The domain is all of the possible values of t that work in the equation. In this case, Becky can sell 0 or more tickets  she can't sell negative tickets. Since we don't know the number of seats in her school auditorium, we can assume that she can theoretically sell an infinite number of tickets. And she can only sell whole tickets; she can't sell 1/2 of a ticket, for example. Therefore, the domain of the function is t = any nonnegative integer.

4Determine the range. The range is the possible amount of money that Becky can make from her sale. You have to work with the domain to find the range. If you know that the domain is any nonnegative integer and that the formula is M(t) = 5t, then you know that you can plug any nonnegative integer into this function to get the output, or the range. For example, if she sells 5 tickets, then M(5) = 5 x 5, or 25 dollars. If she sells 100, then M(100) = 5 x 100, or 500 dollars. Therefore, the range of the function is any nonnegative integer that is a multiple of five.
 That means that any nonnegative integer that is a multiple of five is a possible output for the input of the function.
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Community Q&A

QuestionHow can I find range of a function using limits?Community AnswerIf a function doesn't have a maximum (or a minimum), then you might have to evaluate a limit to find its range. For example, f(x) = 2^x doesn't have a minimum but the limit as x approaches negative infinity is 0, and the limit as x approaches positive infinity is infinity. So the range is (0,infinity) using open intervals because neither limit is ever reached, only approached.

QuestionWhat is AM = GM concept for finding range?Community AnswerThis refers to the Arithmetic Mean (AM)  Geometric Mean (GM) inequality, which states that for positive numbers, the AM is always at least as large as the GM. In some cases, this can be used to find upper or lower bounds for the range of a function. For example, find the range of f(x) = x^2 + 1/x^2. It obviously has a minimum, but where? Many calculus students will immediately take a derivative. This works fine, but if you know the AMGM inequality, there is no need for the heavy artillery of calculus. f(x) = 2 * AM(x^2, 1/x^2). The GM of (x^2, 1/x^2) is 1, and the since the AM is more than that, f(x) is always at least 2, and the range of f is [2, infinity).

QuestionThe function is given that g(x)=x25x+9. How do I find the values of x, which have an image of 15?Community AnswerSimply put g(x) = 15, you will get 2 values of 'x' which satisfy the given quadratic equation. Those values are your answer.

QuestionHow do I find the range of a parabola when it is off of the x or y axis (for instance x=3)?Community AnswerStart by finding the vertex. If the parabola is the form a(xh)^2+k, then (h,k) is the vertex. If it is not in that form but rather in ax^2+bx+c, then get it in the standard form or graph it. From the vertex, if the parabola opens up, then the range will be (k, infinity) and if it opens down the range will be (infinity, k).

QuestionWhat is the range of y=4*3 when the domain is (1,0,2)?Community AnswerSubstitute the elements in the domain to x. The values of y you are getting are the elements of the range.

QuestionHow do I find the range of an equation?DonaganTop AnswererIt's the same as finding the range of a function, as shown above. (This article refers to equations as "functions.")

QuestionIf f(x) = 2x + 4, how can I find the range?Surekha PallemmediCommunity AnswerIf x =1 then f(x) is =6.if x= 2 then f(x) =8 here range is 6,8.
Video
Tips
 For more difficult cases, it may be easier to draw the graph first using the domain (if possible) and then determine the range graphically.Thanks!
 See if you can find the inverse function. The domain of a function's inverse function is equal to that function's range.Thanks!
 Check to see if the function repeats. Any function which repeats along the xaxis will have the same range for the entire function. For instance, f(x) = sin(x) has a range between 1 and 1.Thanks!
References
About This Article
To find the range of a function in math, first write down whatever formula you’re working with. Then, if you’re working with a parabola or any equation where the xcoordinate is squared or raised to an even power, use the formula b divided by 2a to get the x and then ycoordinates. You can skip this step if you’re working with a straight line or any function with a polynomial of an odd number. Next, plug in a few other xcoordinates and solve for their ycoordinates. Finally, plot those points on a graph to see the range of your function. For more on finding the range of a function, including for a relation and in a word problem, scroll down!