Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
29,497
questions
0
votes
0answers
13 views
How to find a function that gives me the indexes of a square matrix
Let a square matrix M with size m, and let i and j, given $0 \leq i = j < m$, so that $a_{ij}$ is a value of the matrix located in the row i and column j (assuming that we start counting with 0 ...
1
vote
1answer
14 views
How to customize a function with a horizontal asymptote?
I'm looking for a function that approaches a y-value as x approaches infinity, something close to: $$y=\frac{x}{x+1}$$ however I have no idea how to customize such a function to suit my needs.
I know ...
0
votes
1answer
26 views
If $\: f:A\to B\:\text{ with }A⊇\{x\},\{y\}\quad$then is $\:f(x)\equiv f(y\mapsto x)\:\text{ & }\:f(y)\equiv f(x\mapsto y)$?
If two variables $x$ and $y$ are both known to be in the domain of a unary function $f$ (but not necessarily defined over the whole domain nor be equal to the other; i.e., $A∋x,y$ while $◇\:\{x\}≠\{y\}...
1
vote
2answers
56 views
When we are computing limit, why the point a(when x approaches a) we plugged in don’t necessarily need to be in the domain of definition
Like this equation :
$$\lim _{x\to 0 }x\sin \frac{1}{x}$$
$x$ is the denominator, but we can nevertheless plug $x=0$ into the function to get the limit, why?
1
vote
1answer
37 views
Even function of a graph with local maximum at x-axis but no local minima
I am having some problems with this question.
Question
Give a possible equation of a function that:
has a single local maximum, located at the x-axis,
has no local minima,
is an even function, and
...
1
vote
0answers
31 views
$f(yf^2(x))=x^3f(xy)$ for all $x,y \in {\mathbb{Q}}^{+}$
Find all functions $f: {\mathbb{Q}}^{+} \rightarrow {\mathbb{Q}}^{+}$ such that:
$f(yf^2(x))=x^3f(xy)$ for all $x,y \in {\mathbb{Q}}^{+}$
My progress:
$P(x;1) \Rightarrow f(f^2(x))=x^3f(x)$
Easily ...
0
votes
1answer
69 views
If a function $f$ is not defined on $[-2,0]$, what can we say about $\lim_{x\to-2}f(x)$?
I was thinking of a strange case in which we can say the limit is undefined. ( I mean different than the limit does not exist.)
So I generated the graph of a piecewise function $f(x)$. There is a ...
-1
votes
0answers
46 views
Comparison of $x^aa^x$ and $ x^x$
I need to compare values of
$x^2a^x$ and $x^x$ at infinity.
approach :
I took f(x) = $(\frac{a}{x})^xx^a$ I found f’(x).
I got information that f is increasing from 0 to a then decreasing after wards.
...
-5
votes
0answers
14 views
What does the graph 3 represents?pls give answer in very detailed manner
enter image description here
This a question from maths contest
0
votes
0answers
48 views
Recursive Definitions and Proposition 2.1.16 in Tao's Analysis I
In his book, Analysis I, Dr. Tao introduces Proposition 2.1.16:
I think I know how to prove that each $a_n$ only takes one value and it is never "redefined," by induction:
As the base case,...
1
vote
0answers
35 views
Is it okay if a function has two different points that have the exact same coordinates?
In math class we are learning about functions.
One of the "stretch" questions is as follows:
"Find the conditions for $a$ and $b$ that make $\{(a,b)(-a,b)(2a,b)(a^2,b)\}$ a function&...
1
vote
1answer
35 views
Why do we have "... for all $x\in\mathbb{R}$" when we define a function after also giving a domain?
I was reading about functions on this wikipedia page and in the section titled 'Notation', they give a proper example definition of a function:
Let $f:\mathbb{R}\to\mathbb{R}$ be the function defined ...
-4
votes
1answer
23 views
Functional equation $\left(f(x)-f(y)\right)f\left(\frac{x+y}2\right)-f(\sqrt{xy}))=0$ [closed]
Select all the continuous functions $f:[0,\infty)\to\mathbb{R}$ that satisfies the following equation
$$\left(f(x)-f(y)\right)f\left(\frac{x+y}2\right)-f\left(\sqrt{xy}\right)=0$$
for each non-...
-3
votes
0answers
11 views
Would it be a closed or open interval about infinity in this function [closed]
If [cot−1 x]+[cos-1 x]=0,
range of the x would be (cos1,1] U (cot1, infinity )or]
The interval including infinity would be a closed one or open one and why
Please a little help here
0
votes
1answer
12 views
Has the associative property been generalized to k-ary functions?
I've been exploring why the associative property is so interesting to mathematicians. Along the way, I have found the rather obvious fact that it only works on binary operations. It needs a concept ...
0
votes
0answers
58 views
If $f(x)=1+4x-x^2$ $\;\forall \; \;x \in R$
If $f(x)=1+4x-x^2$ $\;\forall \; \;x \in R$
$g(x)=\left\{
\begin{array}{ll}
\max f(t); \;x\leq t\leq x+1; \;0\leq x \lt3 \\
{6}; \; \mbox{if } 3\leq x\leq 5
\end{array}
\right.$
Verify ...
0
votes
1answer
71 views
What may be true about the roots of the equation $ax^3 + bx -1 = 0$? (a and b are rational numbers)
Options are as follows
It has 2 irrational and 1 rational root
It has 1 irrational and 2 rational roots
A ll its roots are complex
None of the above
As per the answer should be D.
Let $a = 1, b = 0$
...
1
vote
1answer
97 views
Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $f(x)=f(x^2+1).$
Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $f(x)=f(x^2+1).$
My tried:
If $x \le 1,$ we consider the sequence
$x_0=a\le 1$
$x_{n+1}=x_n^2 +1$
$\...
2
votes
1answer
89 views
Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $f(x-y)+f(y-z)+f(z-x)+27=0.$
Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $$f(x-y)+f(y-z)+f(z-x)+27=0.$$
I actually don't have any ideas to deal with it, but here is some tries:
Let $...
1
vote
0answers
33 views
Submultiplicity when dealing with fractions
I understand that if $f: A \to B$ is a submultiplicative function, then $\exists K : f(ab)\leq K f(a)f(b)$ for every $(a,b)\in A\times A$. Now at some point, by arguing submultiplicity my teacher said ...
-1
votes
1answer
22 views
Need help finding the equation of a cubic polynomial
I need help finding the equation of a real cubic polynomial that cuts the x-axis at 1/2 and -3, has a y-intercept of 30, and passes through (1, -20).
So far I have a(2x-1)(x+3)(bx-c)?
Is there a way ...
0
votes
0answers
41 views
Approximating an exponential function with another function
I have a function $f(\ell)=\exp\Big(-{\frac{a}{\ell b+c}}\Big)$. I would like to rewrite this function in the form $(1-d)g^{\ell}$.
How can I express $g$ and $d$ in terms of $a$, $b$ and $c$?
Here, a,...
0
votes
0answers
21 views
$\frac{\textrm{d}\quad\:}{\textrm{d}\big(x\:\:\text{ s.t. }{x=0}\big)}\Big(|x|\Big):=\text{[undefined]}$. But if $x:∈ℝ\:$→¿◇∃ a ‘best-fit’ derivative?
Let $y=:f(x)\:=|x|\quad\text{s.t.}\:f:ℝ→ℝ$. Under this defined reals-to-reals mapping (graphable on a $ℝ_x×ℝ_y$ cartesian grid, visibly symmetric with window constraints for $y≥0$ centered about $x=0$...
2
votes
3answers
105 views
When is the product of two injective functions, also injective?
I'm currently, by curiosity, investigating when the product of two injective functions, is also injective. My condition is that this is true if and only if there $\nexists x_1, x_2 \in X$ such that $f(...
4
votes
1answer
54 views
extension Rolle's theorem for limit values
Rolle's theorem states that:
If a real-valued function $f$ is continuous on a proper closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f (a) = f (b)$, then there exists at ...
0
votes
1answer
62 views
Find the minimum of $a^4+b^4+c^4+d^4+a^2+b^2+c^2+d^2$.
Let $a,b,c,d$ be real numbers such that $a+b+c+d=0$ and $abcd=1$. Find
the minimum value of $a^4+b^4+c^4+d^4+a^2+b^2+c^2+d^2$.
By $\text{Vieta}$'s theorem, $a,b,c,d$ are the roots of the equation $x^...
0
votes
0answers
28 views
Understanding a question from the topic FUNCTIONS.
I was trying to solve some problems about functions when I got stuck with this problem.
I am unable to understand this. First and foremost,
$$A=[1,2]$$
$$B=[3,4]$$
This seems to be a little daunting. ...
0
votes
1answer
65 views
Finding f(g) from f(x) and g(x)
I'm studying an article where there are two functions:
$e(x) = (e_0/8)\cdot[(2x^3 + x)\sqrt{(1 + x^2)} − \sinh^{−1}(x)]$
$p(x) = (e_0/24)\cdot[(2x^3 - 3x)\sqrt{(1 + x^2)} + 3\sinh^{−1}(x)]$
The ...
0
votes
0answers
22 views
Functional equation $x f(y) -y f(x) =f(\dfrac{y}{x})$ [duplicate]
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R},x\neq 0$ then $$x f(y) -y f(x) =f(\dfrac{y}{x}).$$
My try:
Let $x=1$ then $f(y) -y f(1) = f(y) \Rightarrow f(1) =...
-2
votes
0answers
16 views
0
votes
2answers
42 views
Solve a quartic function with three unknowns and two given roots.
I'd like to know how I could solve the following quartic function: $p(z)=2z^4+az^3+bz^2+cz+3$ given that it $2$ and $i$ should be part of their roots.
I thought I should maybe be trying to turn this ...
0
votes
1answer
41 views
Formal Definition of the Product of Two Sets
I am taking a course in Algebraic Structures, and the notion of product of sets (a.k.a. Caratesian Product) came in. We were given a definition that made me feel I didn't understand it. They gave the ...
0
votes
2answers
38 views
$F(x, y) = \frac{x^2+y^2} {|x|+|y|} $ at $(x, y) \neq (0, 0)$ find if function is continuous or not? [closed]
$F(x, y) = \frac{x^2+y^2}{|x|+|y|}$ at $(x, y) \neq (0, 0)$ find if function is continuous or not?
Continuity and Functions
We have to tell whether function is continuous or not.
I don't know how to ...
2
votes
0answers
29 views
What functions with source and target the rational numbers satisfy the intermediate value theorem?
I am curious to find if there is a characterization of the set $S$ of all functions $f: \mathbb{Q} \to \mathbb{Q}$ where for all intervals $[a,b] \subset \mathbb{Q}$, there exists $x \in [a,b]$ for ...
-5
votes
0answers
35 views
I'm stuck with this proof: [closed]
Proof that -> $\lim\limits_{x\rightarrow 0^+}f(x) = \infty$ if, and only if $\lim\limits_{x\rightarrow \infty}f(\frac{1}{x}) = \infty$.
Thanks everyone :D
1
vote
1answer
74 views
Find the number of injective functions $f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}$ such that $f(i)>f(i+1)$
Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions
$f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}$
such that there exists a unique number
$i\in\{1,2,\ldots,n-1\...
2
votes
2answers
35 views
Doubt proving that $(\forall f,f' \quad g\circ f = g\circ f' \implies f=f' ) \implies g \text{ is injective}$?
I am trying to understand the proof of:
(ii) The function $g : b \to c$ is mono iff the following condition holds: for any two functions $f,f' : a \to b$, $g \circ f = g \circ f'$ implies $f=f'$
...
1
vote
0answers
44 views
Hollow Square Bracket Notation
This function appeared on my HW but I do not recognize the bracket symbols. I am thinking they might be an equivalent notation for absolute value, but I'm not sure and would appreciate any ...
2
votes
2answers
89 views
Looking for function which satisfies $f(n)=f(2n)+f(2n+1)$
I am looking for a function from $\Bbb N^*$ to $\Bbb R^{+*}$
such that $f(n)=f(2n)+f(2n+1)$ (for any $n$ in $\Bbb N^*$).
I also am looking for something smooth, where $f(n+1)-f(n)$ would be strictly ...
1
vote
1answer
44 views
Uniform convergence of a series of functions with arctan
I'm trying to solve the following problem about the uniform convergence of a series of functions involving the function arctan(x).
$$ \sum_{i=1}^\infty \left[1-\frac{n}{x^{2n}}\arctan\left(\frac{x^{2n}...
0
votes
1answer
24 views
How can you determine where two polynomials cross and are tangent to each other, just by factorising the difference?
My textbook asked me to factorise the difference between P(x) and Q(x) and describe the intersection.
I'm having trouble understanding:
Why doing this would allow me to describe the intersection
And ...
0
votes
2answers
30 views
The graph of a polynomial function of degree n is completely determined by any n + 1 points on the curve
Highschool maths student here,
I'm having trouble understanding this statement:
"The graph of a polynomial function of degree n is completely determined by any n + 1 points on the curve"
Any ...
0
votes
0answers
41 views
Finding the value of $\lim_{x\to\infty}\frac{\int_0^x\cos^{-1}(\cos t)dt}{\int_0^x\sqrt{\{t\}-\{t\}^2}dt}$
Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be defined as $f(x)=\cos^{-1}(\cos x)$ and $g(x)=\sqrt{\{x\}-\{x\}^2}$ where $\{x\}$ denotes fractional part of $x$. Find the value of $$\...
2
votes
1answer
46 views
Finding the value of $\displaystyle\int_{x_1+x_2}^{3x_2-x_1}\big\{\frac x4\big\}\left(1+\left[\tan\left(\frac{\{x\}}{1+\{x\}}\right)\right]\right)dx$
If $x_1$ and $x_2$ ($x_1\lt x_2$) are two values of $x$ satisfying the equation $\left|2\left(x^2+\frac1{x^2}\right)+|1-x^2|\right|=4\left(\frac32-2^{x^2-3}-\frac1{2^{x^2+1}}\right)$ then find the ...
3
votes
2answers
72 views
$f:[1,4]\rightarrow[7,14]$ is a concave surjective function then prove that $(f'(x))^2=49/9$ has at least one and at most two roots in $[1,4]$
$f:[1,4]\rightarrow[7,14]$ is a strictly concave surjective function then prove that $(f'(x))^2=49/9$ has at least one and at most two roots in $[1,4]$
$\displaystyle f'(x)=\pm\frac{7}{3}$, so if we ...
-1
votes
0answers
32 views
Integration of unknown functions
Let $ \ f(x) \ $ and $ \ g(x) \ $ be continous positive function such that $ \ f(-x)=g(x)-1 \ $ and $ \ f(x)=g(x)/g(-x) \ . $ Integration of $ \ f(x) \ $ from -20 to 20 is 2020 . Find integration of $ ...
3
votes
2answers
112 views
If $y=f(x)$ is a concave upward function and $y=g(x)$ is a function such that $f'(x)g(x)-g'(x)f(x)=x^4+2x^2+10$ then prove that...
If $y=f(x)$ is a concave upward function and $y=g(x)$ is a function
such that $f'(x)g(x)-g'(x)f(x)=x^4+2x^2+10$ then prove that $g(x)$ has at least one root between consecutive roots of $f(x)=0$
$f'(...
-3
votes
1answer
28 views
Help me with this demonstration involving function composition [closed]
Given two functions $f, g : \mathbb{R}\to\mathbb{R}$, we define the composition of $f$ and $g$, denoted by $f\circ g$, as
$$(f\circ g)(x) := f(g(x))$$
for all $x\in\mathbb{R}$.
Suppose $g$ is ...
2
votes
1answer
38 views
how to obtain the strong convexity inequality
I was playing around with the strong convexity definition and got stuck at some point. I was wondering if someone could kindly help me out.
We say that function $f$ is strongly convex if
$1) f(x) \geq ...
2
votes
3answers
73 views
Using Lagrange multiplier , find minimum value of $xy(x^2 + y^2) +4$ , given that $x^2 + y^2 +xy -1 = 0 , x,y \in \mathbb R$
Using Lagrange multiplier , find minimum value of $$f(x,y)=xy(x^2 + y^2) +4$$ , Given that $g(x,y)=x^2 + y^2 +xy -1 = 0 , $for all values of $ x,y \in \mathbb R$.
My attempt
So i formed a function $$...
