What is the difference between domain and range in mathematics?

Answer 1

The domain of a function is the interval of valid input numbers, and the range is the interval of possible output values for a function. Generally, the domain of a function is all real numbers (often signified by a capital R), because for most functions, any number can be input into it and calculated. This is true for most all polynomial functions, exponential functions, and sine/cosine trigonometric functions.

However, it is easy to recognize when a function will have a restricted domain:

• If a function involves a square root, the inside of the square root cannot be a negative number.
• If a function involves division, the denominator of the division cannot equal zero.
• If a function involves a logarithm, the inside of the logarithm cannot be zero or negative.
• Trigonometric functions are fraught with specific domains, with sine and cosine being the only continuous functions.

These are the three most common things that will signify that a limited domain is present. Also, the inverse trigonometric functions cos-1(x) and sin-1(x) are limited in domain, but this is not so easy to explain. If interested, message me on here and I will explain them or use Wolfram|Alpha to plot these function and investigate them on your own.

For the first type of signification, any square root cannot have a negative number as its argument and remain in the real numbers. So, for instance:

y=sqrt(x)

This function will not exist for any number less than 0 plugged in for x. Therefore, any number from (and including) zero up to positive infinity will result in a valid answer. The domain is therefore [0,infinity), also shown by 0<=x

y=sqrt(x+2)

For this function, x+2 must be greater than or equal to zero. x+2>=0 implies than x>=-2. This means that the domain is [-2,infinity) or -2<=x<=infinity.

For the second type of signification, division by zero is impossible, so any division cannot result in a denominator of zero. Examples:

y=1/x

For this function, x cannot equal zero. So, the domain is all real numbers so long as x != 0 (x does not equal zero).

y=(x4-5x)/(2x3+16)

For this function, 2x3+8 cannot equal zero, So the domain is all real numbers so long as 2x3+16 != 0. This constraint implies 2x3 != -16, which implies x3 != -8, which implies that x != -2. Therefore, the domain of this function is all real numbers so long as x does not equal 2.

y=(x+2)/(x2-9)

Using the same logic as above, the domain of this function is given by x2-9 != 0, which implies x2 != 9, which means x cannot equal 3 or -3. So, its domain is all real numbers so long as x does not equal 3 or -3.

Another type of common signification is logarithmic functions. A logarithmic function cannot exist for zero or negative numbers. This would imply that a number raised to some negative number or zero would yield a negative value, which is not true (any exponential calculation always yields a positive number).

y=ln(x)

For this function, x cannot equal zero or a negative number, so its domain is (0, infinity). This is an open interval (also written as 0

y=log(x2-81)

This function's domain would be given by x2-81>0, which implies x2>81, which implies that any value between x=-9 and x=9 will give an invalid answer. Graphically, the domain of this is all numbers outside of the semicircle of radius 9 centered at the origin.

For trigonometic functions, the fact that division by zero cannot exist often comes into play.

y=tan(x)

For this function, x cannot equal any whole odd multiple of (pi)/2 (3 times (pi)/2, 5 times (pi)/2, 7 times, and so on). This is because any whole odd multiple of (pi)/2 would result in a tangent (which is sine divided by cosine) of 1/0, which is impossible. Therefore, the domain of this function is all real numbers so that x does not equal (n(pi))/2 where "n" is any odd whole number.

y=sec(x)

For this function, x cannot equal any whole multiple of pi. Since secant is the reciprocal of cosine (1/cos(x)), cos(x) cannot equal zero, and this only happens at 0, pi, 2(pi), or any whole multiple of pi. Therefore the domain of this function is all real numbers so that x does not equal n(pi), where "n" is any whole number.

Remember most importantly that domain only concerns x-values. Look for these tell-tale signs can you can find functions whose domains are conspicuous.

The range of a function covers the possible y-values to come from a function. Once again, for many functions this is all real numbers, but again, some tell-tale signs exist:

• Polynomial functions of even degree are restricted to the point that no y-value exist under the lowest vertex of the function (or above the highest vertex if the function opens downward). The most familiar example is a parabola. If a parabola opens upward, no y-values exist below the vertex, and if it opens downward, no values exist above the vertex.
• Exponential functions never yield negative values. For the general function y=ax, any value of x yields a positive number. If the function was vertically shifted, the range would change. Instead of no values above zero, if the function was y=ax-1, no values would exist above -1.
• Semicircular functions are circles, so no values exist over the radius of the semicircle. A true circle cannot be modeled as a function, but a semicircle can. For the function y=sqrt(81-x2), which shows a semicircle of radius 9 above the x-axis, no values exist above 9 or below 0.
• All trigonometirc functions except tangent and cotangent are range-restricted in some manner due to the limits of the unit circle that they are based off of.

y=x2

For this function, the range is all numbers greater than or equal to 0, [0, infinity). The vertex of this parabola is (0,0), so no y-values exist above 0. Its domain would be all real numbers.

y=(4-x)2-3

For this function, the range is all numbers less than or equal to -3, (-infinity, -3]. This parabola opens downward, and its vertex is (-4,-3). Its domain would be all real numbers.

y=ex

For this function, the range is all positive numbers, (0,infinity). It cannot equal zero or any negative numbers. Its domain would be all real numbers.

y=3x+5

For this function, the range is all numbers greater than but not including 5, (5,infinity). 3x cannot equal zero or any negative number, so at the least this function equals just above but not including 5. Its domain would be all real numbers.

y=-sqrt(64-x2)

This function describes the bottom semicircle of a circle of radius 8 centered at the origin. At most, the function can equal 0, and at the least it can equal -8. Therefore, the range of the function is [-8,0]. Its domain is [-8,8].

y=sin(x) or y=cos(x)

For either sine or cosine, the unit circle dictates that the largest value either can obtain is 1 (at x=(pi)/2 for sine and x=0 for cosine), and the smallest value either can obtain is -1 (at x=(3(pi))/2 for sine and x=pi for cosine). Therefore, for both sine and cosine, the range is [-1,1]. Their domains would be all real numbers.

y=23sin(x)

This function's range would simply be 23 times the range of a normal sine range, so it would be [-23,23]. Its domain would still be all real numbers.

y=csc(x)

This function would not have y-values between y=-1 and y=1. Since this is the same as y=1/sin(x), and sin(x) can reach only equal values between -1 and 1, sin(x) would almost always be a fraction unless it was 1 or -1. 1 divided by a proper fraction is a number greater than 1, so there would be no values for csc(x) between -1 and 1. Therefore the range of this function is (-infinity,-1] and [1,infinity).

For a final "test", what is the range and domain of the function:

y=(sqrt(x-4))/(x2-4)+ex

• Because of the square root, x-4>=0, which means x>=4
• Because of the division, x2-4 != 0, which means x cannot equal any -2 or 2
• ex has no impact on the domain

The final domain is [4,infinity).

• By using the domain to analyze the first part of the function, you can realize that the first division part of the function can never equal a negative number. The top can never be negative, so the bottom must be negative for the entire thing to be negative. The bottom cannot be negative, however, because the x-value must be less than 2 for the bottom to be negative, but the domain does not cover these values.
• The first part of the function can at the least equal 0, so this is to say that at the least, this whole function would resemble y=ex+0, which has a minimum of approaching zero.
• Conceivably, there is no maximum to the y-values attainable by the function.

The final range is (0,infinity).