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Concept Version 12
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Domains of Rational and Radical Functions

Rational and radical expressions have restrictions on their domains which can be found algebraically or graphically.  

Learning Objective

  • Calculate the domain of a rational or radical function by finding the values for which it is undefined


Key Points

    • A rational expression is the quotient of two polynomials. It can be expressed as P(x)Q(x)\displaystyle \frac{P(x)}{Q(x)}​Q(x)​​P(x)​​.
    • A rational expression's domain is set such that the denominator cannot equal zero. Therefore, given P(x)Q(x)\displaystyle \frac{P(x)}{Q(x)}​Q(x)​​P(x)​​, Q(x)≠0Q(x)\neq 0Q(x)≠0.
    • To determine the domain of a rational expression, set the denominator equal to zero, then solve for xxx. All values of xxx except for those that satisfy Q(x)=0Q(x)=0Q(x)=0 are the domain of the expression.
    • A radical expression, is expressed as x\sqrt x√​x​​​  and can have other roots other than a square root.  
    • A radical function is expressed as f(x)=xf(x)=\sqrt xf(x)=√​x​​​, (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself.
    • To determine the domain of a radical expression, set the radicand equal to zero, then solve for xxx.  All values of xxx except for those that satisfy x=0\sqrt x=0√​x​​​=0 are the domain of the expression.    

Terms

  • radicand

    The number or expression underneath the radical sign.

  • rational expression

    An expression that can be written as the quotient of two polynomials.


Full Text

Finding Domains of Rational Functions

A rational expression is one which can be written as the ratio of two polynomial functions. Despite being called a rational expression, neither the coefficients of the polynomials nor the values taken by the function are necessarily rational numbers. In the case of one variable, xxx, an expression is called rational if and only if it can be written in the form:

P(x)Q(x)\displaystyle \frac { P(x) }{ Q(x) }​Q(x)​​P(x)​​

where P(x)P(x)P(x) and Q(x)Q(x)Q(x) are polynomial functions in xx x and Q(x)Q(x)Q(x) is not the zero polynomial (Q(x)≠0)(Q(x) \neq 0)(Q(x)≠0).

The domain of a rational expression of is the set of all points for which the denominator is not zero. If the denominator of the equation becomes equal to zero, the expression is undefined at that point.

Example 1:  What is the domain of the rational function:

 f(x)=x3−2x2(x2−5)\displaystyle f(x)= \frac { x^3-2x }{ 2(x^2-5) }f(x)=​2(x​2​​−5)​​x​3​​−2x​​

To find the domain of a rational function, set the denominator equal to zero and solve.  All values of xxx except for those that satisfy 2(x2−5)=02(x^2-5)=02(x​2​​−5)=0 are the domain of the expression.  

2(x2−5)=0\displaystyle 2(x^2-5)=02(x​2​​−5)=0

To solve, divide both sides by 222, add 555 to both sides, and then take the square root of both sides to yield:

x=±5\displaystyle x=\pm \sqrt { 5 }x=±√​5​​​.  

Therefore the domain is the set of all real numbers except the square root of five or negative square root of five.

Notice the graph of the function below.  At the values of x=±5x=\pm \sqrt { 5 }x=±√​5​​​ (which is approximately ±2.2\pm 2.2±2.2), the graph does not exist.

Rational function with restricted domain

The graph of the function: f(x)=x3−2x2(x2−5)f(x)=\displaystyle \frac { x^3-2x }{ 2(x^2-5) }f(x)=​2(x​2​​−5)​​x​3​​−2x​​, where the domain is restricted at x=±5x=\pm \sqrt { 5 }x=±√​5​​​ since the function does not exist at those points.

Example 2:  What is the domain of the rational function: 

f(x)=(x2−2)x\displaystyle f(x)= \frac{\left(x^2-2\right)}{x}f(x)=​x​​(x​2​​−2)​​ 

Algebraically, the domain is the set of all real numbers except zero, since the denominator can not equal zero. One way to determine this is to look at it graphically. We can see that the graph is discontinuous at x=0x=0x=0, indicating that the domain is all numbers other than x=0x=0x=0. This makes sense, because at x=0x=0x=0 we would have to divide by zero, which is undefined.  The red lines of the graph get closer and closer to the value x=0x=0x=0, but never touch.

Rational Function Dividing by xxx

A graph of the equation: f(x)=(x2−2)xf(x)= \frac{\left(x^2-2\right)}{x}f(x)=​x​​(x​2​​−2)​​ . To determine the domain of this function, we can graph it and look for where the function doesn't exist, in this case when x=0x=0x=0.

Finding Domains of Radical Functions

The principal square root function f(x)=xf(x)=\sqrt xf(x)=√​x​​​ (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself.  

Radical Function

The function f(x)=xf(x)=\sqrt xf(x)=√​x​​​ consists of a restricted domain of x≥0x\geq 0x≥0, or non-negative real numbers, since we can not take the square root of a negative number.  

Note that half of the parabola is missing since functions cannot have more than one value at a point, and the square root function is taken to yield a positive value (though (−x)2(-x)^2(−x)​2​​ gives the same value as x2x^2x​2​​ so the square root of a number yyy such that y=x2y=x^2y=x​2​​ would be y=±x\sqrt y = \pm x√​y​​​=±x). It is important to remember when graphing the roots that negative values of xxx will not produce real numbers. This will be explained further in the section on imaginary numbers.

To determine the domain of a radical function algebraically, find the values of xxx for which the radicand is nonnegative (set it equal to ≥0\geq 0≥0) and then solve for xxx.  The radicand is the number or expression underneath the radical sign.  All values of xxx except for those that satisfy x≥0\sqrt x \geq 0√​x​​​≥0 are the domain of the function. 

Example 3:  What is the domain of the radical function:

 f(x)=x−3+4\displaystyle f(x) = \sqrt {x-3} +4f(x)=√​x−3​​​+4

Set the radicand greater than or equal to zero and solve for xxx to find the restrictions on the domain:

 x−3≥0\displaystyle {x-3} \geq 0x−3≥0

Therefore x≥3x \geq 3x≥3.  So, all real number greater than or equal to 333 is the domain of the function.

Radical function

The graph of the equation: f(x)=x−3+4f(x) = \sqrt {x-3} +4f(x)=√​x−3​​​+4.  The function has the domain of all real numbers greater than or equal to 333, as shown in the graph above.  

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