In the example, we need to solve for \(x\): Normally, if possible, we should prefer the analytical/algebraic way. The "graphical method" to find the range has that problem: it is appealing from an intuitive point of view, but it is rather thin in terms of content. Such analysis is correct in terms of the result, but it is flimsy in terms of the reasoning. And then, the conclusion is that the range is the whole real line, which is \((-\infty, +\infty)\) using interval notation. The intuition is that function can take as negative and as positive as we want values, by selecting large enough (positive or negative) \(x\) values. And analogously, when \(x\) is very negative, the value of the function is also very negative. Moreover, when \(x\) is large and positive, the value of the function is also large and positive. The graph above does not show any minimum or maximum points. Other Strategies for Finding Range of a functionĪs we saw in the previous example, sometimes we can find the range of a function by just looking at its graph.įor example, say you want to find the range of the function \(f(x) = x + 3\). The risk of using the graph to find the range is that you could potentially misread the critical points in the graph and give an inaccurate evaluation of the where the function reaches its maximum or minimum. We can see that, based on the graph, the minimum is reached at \(x = 2\), which is exactly what was found to the x-coordinate of the vertex. The graph of the function \(f(x) = x^2 - 4x + 3\) makes it even more clear: Since the minimum value reached by the parabola is \(-1\), we conclude that the range is \([-1, +\infty)\), which is the same conclusion as the one found algebraically. Now, the y-coordinate of the vertex is simply found by plugging the value \(x_V = 2\) into the quadratic function:
RANGE MATH HOW TO
So this is the algebraic way, the way how to find range of a function without graphing.įind the range of the function \(\displaystyle f(x) = \frac = 2\] Of course, that could be hard to do, depending on the structure of the function \(f(x)\), but its what you need to do. We need to determine for which values of \(y\) the above equation can be solved for \(x\). Then, we will consider a generic real number \(y\) and we will try to solve for \(x\) the following equation: Say that we need to get the range of a given function \(f(x)\). Yet, there is one algebraic technique that will always be used. Same as for when we learned how to compute the domain, there is not one recipe to find the range, it really depends on the structure of the function \(f(x)\). The Algebraic Way of Finding the Range of a Function
For a more conceptual approach to domain and range, you can In this tutorial we will concentrate more on the mechanics of finding the range. So in other words, we need to find \(x\) so that \(q(x) = b\), which is another way of asking whether or not \(b\) is in the range of the function \(q(x)\). We would like to know how many input units are needed to produce \(b\) units of output. For example, you may have a production function \(q(x)\), which gives you the amount of output obtained for \(x\) units of input.
The task of finding what points can be reached by a function is a very useful one. Or in other words, it allows you to find the set of all the images via the function Learning how to find the range of a function can prove to be very important in Algebra and Calculus, because it gives you the capability to assess what values are reached by a function.
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