Range of a Function
Date: 01/05/97 at 09:13:29 From: Anonymous Subject: Functions Dear Dr. Math, Please could you tell me how to find the range of a given function? For example: The functions f and g are defined over the set of real numbers by f : x --> 3x-5 g : x --> e^-2x a) State the range of g Thank you for your help in this matter. Regards, Tracy Pritchard
Date: 01/05/97 at 17:26:26 From: Doctor Pete Subject: Re: Functions Hi, First, as a side note, it is not a common practice to define functions as in the above. For instance, most people would write g as follows: g : R --> R, g(x) = e^(-2x), where R denotes the set of real numbers. The first part tells us what type of mapping g is. Generally speaking, if we write: f : U --> V then f is a function that maps elements from the set U to elements in the set V. U is called the domain, and V is called the codomain, or range. It is often not necessary to be completely specific when specifying the codomain; for instance, in the function: f : R --> R, f(x) = x^2 + 1 we could have written f : R --> R+ where R+ denotes the set of all positive reals. This is because x^2 is always greater than or equal to 0 for all real x so there is no way that this function can produce a negative number when the domain consists of only real numbers. On the other hand: f : C --> C, f(x) = x^2 + 1 where C is the complex numbers, one cannot make such a restriction on the codomain because the domain has been specified to be the set of all complex numbers. With this in mind, we should write your problem as follows: If f : R --> U, f(x) = 3x-5, g : R --> V, g(x) = e^(-2x) specify U and V. Clearly, U and V are subsets of R (though this is not always the case). We may find V informally by graphing it. However, of particular interest is the fact that the function e^x > 0 for all real x. In other words: h : R --> R+, h(x) = e^x Also, note that: k : R --> R, k(x) = -2x It is clear that the range of k is the same as its domain (graph it). Finally, it should be observed that: g = h o k, or g(x) = h(k(x)) That is, g is the composition of h and k. We can then make the following diagram: k R --->R \ | \ | g \ | h \ | \| R+ So we see that the range of g is R+, the set of positive reals. In general it is not always easy to completely specify the codomain of a function. Also, it is usually necessary to specify the domain in order to determine the restrictions on the codomain, as I have mentioned above. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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