# What is domain and range: Definition and 71 Discussions

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is most often denoted by letters such as f, g and h, and the value of a function f at an element x of its domain is denoted by f(x); the numerical value resulting from the function evaluation at a particular input value is denoted by replacing x with this value; for example, the value of f at x = 4 is denoted by f(4). When the function is not named and is represented by an expression E, the value of the function at, say, x = 4 may be denoted by E|x=4. For example, the value at 4 of the function that maps x to

(
x
+
1

)

2

{\displaystyle (x+1)^{2}}
may be denoted by

(
x
+
1

)

2

|

x
=
4

{\displaystyle \left.(x+1)^{2}\right\vert _{x=4}}
(which results in 25).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.
Functions are widely used in science, engineering, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.

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1. ### Finding the domain (and range) of a logarithmic function

Problem statement : I copy and paste the problem from the text. You will note that I added the range myself, because it seemed relevant and yet I couldn't do it. Attempt : I could evaluate the domain. The base of the function ##x-4>0\Rightarrow x>4.\hspace{60 pt} (1)## The function...

16. ### MHB Domain and range of inverse functions (circular and hyperbolic)

I've always been having trouble with the domain and range of inverse trigonometric functions. For example, let's start with an easy one: $\sin^{-1}\left({x}\right)$ Process: First, I draw out the function of $\sin\left({x}\right)$. Then I look at its range and attempt to restrict it so that it...
17. ### Identifying Domain and Range of Ovals with Given Points

Homework Statement There are two vertical ovals, one containing the x-coordinate values and one containing the y-coordinate values. The x-axis values: -1, 8, 0 and the y-axis values: -4, 0, -1 (-1,-4) (8,0) and (0,-1) are the three points. Homework Equations I'm not sure this is an equation...
18. ### A Question about the Domain and Range of a Function

Homework Statement I constructed my own question to try to make sense of the following notation. g(x) = 2√x g : X → Y. What does X and Y equal? Homework Equations For 2√x, x = or > than 0. The Attempt at a Solution g(x) = 2√x g : [0, ∞) → [0, ∞) So X = Y = [0, ∞) The...
19. ### MHB Find the constants given the domain and range

Suppose you have a function y = f(x) such that the domain of f(x) is 1 ≤ x ≤ 6 and the range of f(x) is −3 ≤ y ≤ 5. a) Find constants B and C so that the domain of f(B(x − C)) is 8 ≤ x ≤ 9 B= C= b) Find constants A and D so that the range of Af(x) + D is 0 ≤ y ≤ 1 A= D= I'm working on...
20. ### Domain and Range of this function

Homework Statement What is the domain and range of the function y = \frac{2}{x^2-4} The Attempt at a Solution I'm lost, and I noticed it seems like a difference of squares so what I did is y = \frac{2}{(x-2)(x+2)}. So the x value of vertex is 0. And i subbed this into the equation...
21. ### MHB Mr.Ask's question at Yahoo Answers (Two variables, domain and range)

Here is the question: Here is a link to the question: Find the range and domain of function of 2 varibles:? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
22. ### MHB Myles's question at Yahoo Answers (Domain and range)

Here is the question: Here is a link to the question: Find the domain and range of f(x)= (e^x)/(1+2e^x)? - Yahoo! AnswersI have posted a link there to this topic so the OP can find my response.
23. ### Determine the domain and range of the function y=5−e^(−x/2).

Homework Statement Determine the domain and range of the function y=5−e^(−x/2). I am having some trouble with this. Does any have some tips to get started, or a good link to a instructional video? My textbook does not have this section :( Thanks all! Homework Equations The...
24. ### Domain and Range of a function

Homework Statement If the maximum allowable width, w, of the wall is 100 cm, what are the domain and rand of the function? Remember to include units for both the domain and range. Homework Equations w(d)=2d w(d)=100, so 100=2d, there fore d=100/2 The Attempt at a Solution...
25. ### Domain and Range of A Function

I have a question about domain and range.What are the domain and range of cos(e^{-×}) and \frac{\left|2×-1\right|}{sin(\frac{1}{2}\pi×-\pi)} ?? Thanks..
26. ### Finding the domain and range without a calculator

Homework Statement Find the domain and range of the following function without the use of a calculator: f(x) = sec (pi x/4) Homework Equations As far as I know, this problem doesn't specifically require "equations". Therefore I am leaving this section blank. Not because I am a...
27. ### Domain and range of complex functions

If I am given a function: f(z) = w(x,y) = u(x,y) + iv(x,y) How do I find the domain and range of that function? Are there any good lectures online on this? I know I have to use two planes; One for u and v, and one for x and y. Besides that, I am lost.
28. ### Definition of function domain and range

I am a bit confused about this matter. While I was studying Calculus I saw an excercise like this: The domain of f(x) [0,2] and the range is [0,1], it also shows its graphic, though it is not important it is something like a parabola, its maximum point is (1,1) and its intersection points are...
29. ### Domain and range of f(x) = x/lxl

Homework Statement Find the domain and range of the function: F(x) = \frac{x}{lxl} Homework Equations ? The Attempt at a Solution domain is all x such that x ≠0 range: do i have to make this into a piecewise function, if so, how?
30. ### Determining Domain and Range of Cartesian Equations

Homework Statement Eliminate the parameter to find the Cartesian equation of the curve. Problem #1: x = sin t, y = csc t, 0 < t < ∏ / 2 Problem #2: x = sin θ, y = cos 2θ Homework Equations What's shown above is what's listed in the book. However, the authors felt compelled to...
31. ### What is the domain and range of y=-0.7x+1.9 and y=0.7x+1.9?

Homework Statement State the domain and range of the function that models each half of the roof (triangle) Homework Equations Y= -0.7x+1.9 (left half) Y= 0.7x+1.9 (right half) The Attempt at a Solution Why is the answer: Left half: domain: −2.7≤x≤0 range: 0≤y≤1.9 Right...
32. ### Domain and range of the function (arctan(ln(sqrtx)-1)))^3

Homework Statement f(x) = (arctan(ln(sqrtx - 1)))^3 Homework Equations domain of arctan: all real numbers range of arctan: -∏/2, ∏/2 The Attempt at a Solution I know that domain is x>0 when x ≠ 1, because I need a positive number to go under the radical and the natural log of 0 is...
33. ### Domain and range of composite function

Homework Statement The function f has domain (-∞, ∞) and is defined by f(x) = 3e2x. The function g has domain (0, ∞) and is defined by g(x) = ln 4x. (a) Write down the domain and range of f∘g. (b) Solve the equation (f∘g)(x) = 12 2. The attempt at a solution (a) Is it correct...
34. ### Challenge problem to the community - - - domain and range

What are the domain and range of the following: (Note: It is a relation that is not a function.) y^2(x^2 - 1) = x^4
35. ### What are the domain and range of this relation

in interval notation? y^2(x^2 - 1) = x^4 (This is my own problem.)
36. ### Graphing a function and finding the domain and range

Homework Statement y= -4(√4x-1) -1 in case it's not clear, the 4x-1 part is all under the radical the question is to graph and find the domain and range Homework Equations The Attempt at a Solution This is probably what your supposed to do , but not sure, OK To get the domain set everything...
37. ### Domain & Range of f(x)=-√(t(1-t))

f(x) = -√(t(1-t)) Domain: 0≤t≥1 (from Null Factor law) Range: 0≤t However, the range was wrong. The answers said that it was -0.5≤t≥0. I have no idea where the -0.5 came from. I substituted the domain values in but i didn't work. It just came out with zero. Can anyone suggest any ideas...
38. ### Find functions with given domain and range

1) Find a continuous function f: (0,1)->R with f[(0,1)]=R I couldn't think of any function except tangent, but its domain is NOT (0,1) though? Any suggestions? 2) Find a continuous function f: (0,1)->R with f[(0,1)]=[0,1] I couldn't think of any function that I know. Any suggestions? 3) Find a...
39. ### Domain and Range of f(x,y)=2x^2+3x^2+1: Real Numbers and Minimum Value of 1

Homework Statement Find the domain and the range f(x,y)=2x2+3x2+1 Homework Equations non The Attempt at a Solution ... i said that x,y are all real #'s and f(x,y)>=1 or z>=1 is there anything wrong with what i has attempted at? if do could you explain... thanks in advance
40. ### What is the relationship between the domain and range of a function?

Could someone please explain to me what the range and domain of a function is and how to find them? I just don't understand what they mean Thank You
41. ### Domain and range can't be negative

Homework Statement Given f:A→B and g:B→C, let h=g(f(a)) If h is injective, then g is injective. Give a counter example. Homework Equations Injection: Let f:A→B For all f(x1)=f(x2) implies x1=x2 The Attempt at a Solution f(a)=\sqrt{a} from [0, infinity]→[0,infinity]...
42. ### Domain and range of a function help

So I'm skipping pre-cal with full intentions of picking up on AP Calc. Our summer course work is taken from a book called preparing for calculus. After giving me some who-dad that I knew already everything changed in an instant. There's a little gap that isn't quiet filled. A function is...
43. ### Which concepts in algebra/trig. have a domain and range

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...
44. ### Domain and Range of fg(x) for f(x)= e^{2x} + 3 and g(x) = ln(x-1)

Homework Statement Find the range for the function; fg(x) and state its range where f(x)= e^{2x} + 3 and g(x) = ln(x-1) The Attempt at a Solution Putting g into f I reach; fg(x) = e^{2ln(x-1)}+3 (1) - here is where I have the problem, now, I know that it simplifies down to fg(x)...
45. ### Parabolas: Domain, Range, and Completing the Square

Homework Statement For the following questions, find the Domain and Range. Use either the graph or turning point method. Homework Equations 1. y= -x+6x-8 2. y= x^2-3x-5 3. y= -x^2+4x+4 And a question on the matter, when you complete the square, as in when I half the middle term...
46. ### Finding Domain and Range of Functions

Find the domain and range of y=sin^2x I have a hard type computing this into my graphing calculator. Can someone help in the steps to find the domain?
47. ### What is the Difference Between a Vector Space and a Coordinate Space?

Say we have a function, f(x)=x^3 one would say "f is a function from \mathbb{R} to \mathbb{R}" or f: \mathbb{R}\to\mathbb{R} Then say we have a vector function, \vec{g}(t)=<t^2+1,t> . How would one use the above notation? Would it be \vec{g}: \mathbb{R}\to\mathbb{R}^2 ? And...
48. ### Restricting Domain and Range in broken conics

Homework Statement How do you know whether you restrict the domain or range with a horizontal ellipse ? Homework Equations x^2/49 +y^2/10=1 The Attempt at a Solution
49. ### Domain and range of the function help needed work shown

domain and range of the function help needed.. work shown! Given the function f(x)= (3x+4)/(5-2x) find the domain and range of the inverse function of f(x) work shown: it was proven to be one to one since f(x1)= f(x2) found inverse f(x) to be (5x-4)/(3+2x), i believe this is correct...
50. ### What is the natural domain and range for f(x,y) = 1/sqrt(x^2-y)?

define the natural domain and range for f(x,y) = 1/sqrt(x^2-y) I get D= {(x,y)|x^2-y>0} same as saying y can not equal x^2 R= {(x,Y)|f>0} can someone tell me if I've approached this correctly?