Which concepts in algebra/trig. have a domain and range

[land_of_ice]

In algebra/trig. what kinds of problems have a domain and a range ?
Is this true for anything that can be graphed? Because a lot of the time after you finish a problem in precalc. it then asks for the domain or range, so the question becomes, in mathematics, is it possible to find the domain and range, for ANY math problem, or what kind of areas of math does the concept of domain and range apply to ? To any math problem period?

 

[LumenPlacidum]

All functions have domains and ranges. Functions are items to which you give them a value and they give you a value (usually these are numbers).

For example, all of the following are functions, and as such they have domains and ranges (which are also given, based on the typical real number analysis concepts):
f(x) = x (D = all real numbers, R = all real numbers)
f(x) = x+3 (D = all real numbers, R = all real numbers)
f(x) = 5x (D = all real numbers, R = all real numbers)
f(x) = 1/x (D = all real numbers except x=0, R = all real numbers except 0)
f(x) = x^2 (D = all real numbers, R = all non-negative real numbers)
f(x) = sqrt(x) (D = all non-negative real numbers, R = all non-negative real numbers)
f(x) = e^x (D = all real numbers, R = all positive real numbers)
f(x) = log(x) (D = all positive real numbers, R = all real numbers)
f(x) = sin(x) (D = all real numbers, R = the inclusive interval from -1 to 1)
f(x) = cos(x) (D = all real numbers, R = the inclusive interval from -1 to 1)
f(x) = tan(x) (D = all real numbers except those of the form n*pi/2 for n being an integer, R = all real numbers)
f(x) = arcsin(x) (D = the inclusive interval from -1 to 1, R = the inclusive interval from -pi/2 to pi/2)
f(x) = arccos(x) (D = the inclusive interval from -1 to 1, R = the inclusive interval from 0 to pi)

Only functions have domains and ranges because that's where their concepts make sense. For example, the domain of f(x) = x^2 is all the real numbers and the range is all the non-negative real numbers. That means that I can feed this function a real number (like, for example, 3) and it will spit out a non-negative real number (in this case, 3^2 = 9). I can't expect to get -3 out from this function no matter what I give it.

If I were talking about the function f(x) = arcsin(x), then the domain is the inclusive interval from -1 to 1 and the range is the inclusive interval from -pi/2 to pi/2. This means that I can feed it only a value between -1 and 1 (inclusive), like 0.5, and it will give me a response somewhere between -pi/2 and pi/2, in this case, pi/6.

 

[HallsofIvy]

No, not only functions. Relations also have "domain" and "range".

 

[Martin Rattigan]

...
f(x) = tan(x) (D = all real numbers except those of the form n*pi/2 for n being an integer, R = all real numbers)
...

A minor correction.

f(x) = tan(x) (D = all real numbers except those of the form (2n+1)*pi/2 for n being an integer, R = all real numbers)

 

[CellCoree]

In algebra and trigonometry, the concepts of domain and range are essential in understanding and solving various problems. The domain refers to the set of values that a function can take as inputs, while the range represents the set of values that a function can produce as outputs. These concepts are applicable to any mathematical problem that involves functions, whether it can be graphed or not.

In algebra, problems involving linear, quadratic, exponential, and logarithmic functions all have a domain and range. For example, in a linear function, the domain would be all real numbers, while the range would depend on the slope and y-intercept of the line. In trigonometry, the domain and range are essential in understanding the behavior of trigonometric functions such as sine, cosine, and tangent.

It is important to note that the concept of domain and range is not limited to functions that can be graphed. Any mathematical problem that involves input and output values can have a domain and range. For instance, in statistics, the domain and range can be used to determine the minimum and maximum values of a data set.

In conclusion, the concept of domain and range is applicable to various areas of mathematics and is not limited to problems that can be graphed. It is a fundamental concept that helps us understand the behavior of functions and solve mathematical problems.