Finding Domain and RangeLearning Objective(s)Find the domain of a square root function.Find the domain and range of a function from the algebraic form.IntroductionFunctions are a correspondence between two sets, called thedomainand therange. When defining a function, you usually state what kind of numbers the domain (x) and range (f(x)) values can be. But even if you say they are real numbers, that doesnt mean thatallreal numbers can be used forx. It also doesnt mean that all real numbers can be function values,f(x). There may be restrictions on the domain and range. The restrictions partly depend on thetypeof function.In this topic, all functions will be restricted to real number values. That is, only real numbers can be used in the domain, and only real numbers can be in the range.Restricting the domainThere are two main reasons why domains are restricted.You cant divide by 0.You cant take the square (or other even) root of a negative number, as the result will not be a real number.In what kind of functions would these two issues occur?Division by 0could happen whenever the function has a variable in thedenominatorof a rational expression. That is, its something to look for inrational functions.Look at these examples, and note that division by 0 doesnt necessarily mean thatxis 0!FunctionNotes

Ifx= 0, you would be dividing by 0, sox 0.

Ifx= 3, you would be dividing by 0, sox 3.

Although you can simplify this function tof(x) = 2, whenx= 1 the original function would include division by 0. Sox 1.

Bothx= 1 andx= 1 would make the denominator 0. Again, this function can be simplified to, but whenx= 1 orx= 1 theoriginalfunction would include division by 0, sox 1 andx 1.

This is an example withnodomainrestrictions, even though there is a variable in the denominator. Sincex2 0,x2+ 1 can never be 0. The least it can be is 1, so there is no danger of division by 0.

Square roots of negative numberscould happen whenever the function has a variable under a radical with an even root. Look at these examples, and note that square root of a negative variable doesnt necessarily mean that the value under the radical sign is negative! For example, ifx= 4, then x= (4) = 4, a positive number.FunctionRestrictions to the Domain

Ifx< 0, you would be taking the square root of a negative number, sox 0.

Ifx< 10, you would be taking the square root of a negative number, sox10.

When is -xnegative? Only whenxis positive. (For example, ifx=3, thenx= 3. Ifx= 1, thenx=1.) This meansx 0.

x2 1 must be positive,x2 1 > 0.Sox2 > 1. This happens only whenxis greater than 1 or less than1:x1 orx 1.

There arenodomainrestrictions, even though there is a variable under the radical. Sincex2 0,x2+ 10 can never be negative. The least it can be is 10, so there is no danger of taking the square root of a negative number.

Domains can be restricted if:the function is a rational function and the denominator is 0 for some value or values ofx.the function is a radical function with an even index (such as a square root), and the radicand can be negative for some value or values ofx.

RangeRemember, here the range is restricted to all real numbers. The range is also determined by the function and the domain. Consider these graphs, and think about what values ofyare possible, and what values (if any) are not. In each case, the functions are real-valuedthat is,xandf(x) can only be real numbers.Quadratic function,f(x) =x2 2x 3

Remember the basic quadratic function:f(x) =x2must always be positive, sof(x) 0 in that case. In general, quadratic functions always have a point with a maximum or greatest value (if it opens down) or a minimum or least value (it if opens up, like the one above). That means the range of a quadratic function will always be restricted to being above the minimum value or below the maximum value. For the function above, the range isf(x) 4.Other polynomial functions with even degrees will have similar range restrictions. Polynomial functions withodddegrees, likef(x) =x3, will not have restrictions.Radical function,f(x)=

Square root functions look like half of a parabola, turned on its side. The fact that the square root portion must always be positive restricts the range of the basic function,, to only positive values. Changes to that function, such as the negative in front of the radical or the subtraction of 2, can change the range. The range of the function above isf(x) 2.Rational function,f(x) =

Rational functions may seem tricky. There is nothing in the function that obviously restricts the range. However, rational functions haveasymptoteslines that the graph will getcloseto, but never cross or even touch. As you can see in the graph above, the domain restriction provides one asymptote,x= 6. The other is the liney= 1, which provides a restriction to the range. In this case, there are no values ofxfor whichf(x) = 1. So, the range for this function is all real numbers except 1.Determining Domain and RangeFinding domain and range of different functions is often a matter of asking yourself, what values can this functionnothave?Example

ProblemWhat are the domain and range of the real-valued functionf(x) =x+ 3?

This is alinearfunction. Remember that linear functions are lines that continue forever in each direction.

Any real number can be substituted forxand get a meaningful output. Foranyreal number, you can always find anxvalue that gives you that number for the output. Unless a linear function is a constant, such asf(x) = 2, there is no restriction on the range.

AnswerThe domain and range are all real numbers.

Example

ProblemWhat are the domain and range of the real-valued functionf(x) = 3x2+ 6x+ 1?

This is aquadraticfunction. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used forxto get a meaningful output.Because the coefficient ofx2is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value.

Thevertex, or turning point, is at (1, 4). From the graph, you can see thatf(x) 4.

AnswerThe domain is all real numbers, and the range is all real numbersf(x) such thatf(x) 4.

You can check that the vertex is indeed at (1, 4). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the sameyvalue. The vertex must lie on the line of reflection, because its the only point that does not have a mirror image!In the previous example, notice that whenx= 2 and whenx= 0, the function value is 1. (You can verify this by evaluatingf(2) andf(0).) That is, both (2, 1) and (0, 1) are on the graph. The line of reflection here isx= 1, so the vertex must be at the point (1,f(1)). Evaluatingf(1)givesf(1)=4, so the vertex is at (1, 4).Example

ProblemWhat are the domain and range of the real-valued function?

This is aradicalfunction. The domain of a radical function is anyxvalue for which the radicand (the value under the radical sign) is not negative. That meansx+ 5 0, sox 5.Since the square root must always be positive or 0,. That means.

AnswerThe domain is all real numbersxwherex 5, and the range is all real numbersf(x) such thatf(x) 2.

Example

ProblemWhat are the domain and range of the real-valued function?

This is arationalfunction. The domain of a rational function is restricted where the denominator is 0. In this case,x+ 2 is the denominator, and this is 0 only whenx= 2.For the range, create a graph using a graphing utility and look for asymptotes:

One asymptote, a vertical asymptote, is atx=2, as you should expect from the domain restriction. The other, a horizontal asymptote, appears to be aroundy =3. (In fact, it is indeedy= 3.)

AnswerThe domain is all real numbers except 2, and the range is all real numbers except 3.

You can check the horizontal asymptote,y= 3. Is it possible forto be equal to 3? Write an equation and try to solve it.

Since the attempt to solve ends with a false statement0 cannot be equal to 6!the equation has no solution. There is no value ofxfor which, so this proves that the range is restricted.Find the domain and range of the real-valued functionf(x) =x2+ 7.A) The domain is all real numbers and the range is all real numbersf(x) such thatf(x)7.B) The domain is all real numbersxsuch thatx 0 and the range is all real numbersf(x) such thatf(x) 7.C) The domain is all real numbersxsuch thatx 0 and the range is all real numbers.D) The domain and range are all real numbers.Show/Hide Answer

SummaryAlthough a function may be given as real valued, it may be that the function has restrictions to its domain and range. There may be some real numbers that cant be part of the domain or part of the range. This is particularly true with rational and radical functions, which can have restrictions to domain, range, or both. Other functions, such as quadratic functions and polynomial functions of even degree, also can have restrictions to their range.