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 Ex=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.
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 ##x4>0\Rightarrow x>4.\hspace{60 pt} (1)##
The function...
$$y = f(x) = \sqrt{9x^2}$$
According to me,
Domain: $$ 9x^2 \geq 0 \implies (x+3)(x3) \leq 0 \implies x \in [3,3] $$
which is correct, but this is how I calculate the range:
$$y = \sqrt{9x^2} \implies y^2 = 9x^2 \implies x^2 = 9y^2$$
Now, since $$ 9x^2 \geq 0 $$
we get $$99+y^2 \geq 0...
Hi everyone
I have the solution to this question, but I'm not sure I understand it.
Why is the domain of the composite function and not [0, pi]?
Is it because tan^1 (0 and R+) will always give a value between (pi/2, pi/2)? I.e. the domain of the composite function refers to x.
Is 0...
The domain and range of this function will be the same.
We can let ##𝑓(𝑥)=\sqrt{x},𝑥≥0##
However, ##𝑦=𝑓(𝑥)≥0##, so the domain and range of ##f## are ##[0,+∞)##
And since ##f## is a function, ##f^{1}s## domain is the range of ##f## and ##f^{1}s## range is ##f’s## domain.
In other words...
Find the inverse function $f^{1}$ of $f$.
Find the range of $f$ and the domain and range of $f^{1}$
$$f(x)=5 \sin x +2; \quad \frac{\pi}{2}\le x \le \frac{\pi}{2}$$
exchange x for y and solve for y
\begin{align*}\displaystyle
x&=5\sin y + 2 \\...
Homework Statement
Consider the function ##f\left(x\right)=\sqrt {x+2}##. Determine if the function is a onetoone function, If so, find ##f^{1}\left(x\right)## and state the domain and range of ##f\left(x\right)## and ##f^{1}\left(x\right)##
Homework Equations
N/A
The Attempt at a...
Homework Statement
f(x,y) = 1/y^2x
find the domain of f.
Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f.
Homework Equations
I know that the domain of the function is anywhere that the function is defined.
The Attempt at a Solution
in the case of...
Hello, I'm sorry for registering to the the boards and immediately posting a new thread. I did use the search function to try and find a similar thread that addresses the issue I'm having but nothing jumped out at me and frankly I'm so green that I wouldn't know what to look for. This is my...
Homework Statement
Suppose f is a function with domain [2,10] and range [5,10]. Find the domain and range of the following functions.
(a) f(2x+4)
(b) 2f(x)+4
The Attempt at a Solution
[/B]
Would I just substitute the in the domain and range values to find the answer?
Homework Statement
Find the domain and range of the following function:
f(x, y) = ln(x+y)
Homework EquationsThe Attempt at a Solution
I know that the natural log ln(x) is only defined when x>0, so does that mean that ln(x+y) is only defined when x+y>0? Also, would the range just be all positive...
Homework Statement
Specify the domain and range of f(x, y) = arccos(y − x2). Indicate whether the domain is (i)
open or closed, and (ii) bounded or unbounded. Give a clear reason in each case.Homework EquationsThe Attempt at a Solution
yx2 >= 1
y >= x2 1
yx2 <= 1
y <= x2 +1
I sketched it...
I was thinking about a function whose domain and range are absolutely one point , then I'm stuck in finding the domain and range of the following function .. could anyone help
$$y=\sqrt{x^2+y^2}$$
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...
Homework Statement
There are two vertical ovals, one containing the xcoordinate values and one containing the ycoordinate values. The xaxis values: 1, 8, 0 and the yaxis 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...
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...
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...
Homework Statement
What is the domain and range of the function y = \frac{2}{x^24}
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}{(x2)(x+2)}. So the x value of vertex is 0. And i subbed this into the equation...
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.
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.
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...
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...
I have a question about domain and range.What are the domain and range of cos(e^{×}) and \frac{\left2×1\right}{sin(\frac{1}{2}\pi×\pi)} ??
Thanks..
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...
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.
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...
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?
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...
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...
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...
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...
Homework Statement
y= 4(√4x1) 1
in case it's not clear, the 4x1 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...
f(x) = √(t(1t))
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...
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...
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
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]...
So I'm skipping precal 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 whodad that I knew already everything changed in an instant. There's a little gap that isn't quiet filled.
A function is...
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...
Homework Statement
Find the range for the function;
fg(x) and state its range where
f(x)= e^{2x} + 3 and g(x) = ln(x1)
The Attempt at a Solution
Putting g into f I reach;
fg(x) = e^{2ln(x1)}+3 (1)  here is where I have the problem, now, I know that it simplifies down to
fg(x)...
Homework Statement
For the following questions, find the Domain and Range.
Use either the graph or turning point method.
Homework Equations
1. y= x+6x8
2. y= x^23x5
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...
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...
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
domain and range of the function help needed.. work shown!
Given the function f(x)= (3x+4)/(52x) 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 (5x4)/(3+2x), i believe this is correct...
define the natural domain and range for f(x,y) = 1/sqrt(x^2y)
I get D= {(x,y)x^2y>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?