What is Piecewise function: Definition and 103 Discussions
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself.
A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example.
I understand that I can divide this shape into a few parallelograms and a triangle and calculate the center of mass of each, but am confused as to what I should do after that. My physics teacher also wants us to use integrals, but I'm assuming I can calculate the COM of each parallelogram and...
I am trying to write a python script to plot the function,
Where
##V_0 = 5~V##
##t_0 = 10~ms##
##\tau = 5~ms##
My script that I have written to try to do this is,
Which plots,
However, the plot is meant to look like this with the horizontal line.
Can someone please give me some guidance to...
Let's say I want to describe a massive box in spacetime as described by the Einstein Field Equations. If one were to construct a metric in cartesian coordinates from the Minkowski metric, would it be reasonable to use a piecewise Stress-Energy Tensor to find our metric? (For example, having...
Use the graph to investigate
(a) lim of f(x) as x→2 from the left side.
(b) lim of f(x) as x→2 from the right side.
(c) lim of f(x) as x→2.
Question 20
For part (a), as I travel along on the x-axis coming from the left, the graph reaches a height of 4. The limit is 4. It does not matter...
My apologies. I posted the correct problem with the wrong set of instructions. It it a typo at my end. Here is the correct set of instructions for 28:
Use the graph to investigate limit of f(x) as x→c. If the limit does not exist, explain why.
For (a), the limit is 1.
For (b), the limit DOES...
I have to find:
g(1)=
and
g(5)=
I have drawn the graph and I am a little unsure where to go from there. I know area is involved somehow but not entirely sure what to do. Any help is appreciated
If the function is not differentiable at point. Can we consider this point is critical point to the function?
f(x) = (x-3)^2 when x>0
= (x+3)^2 when x<0
he asked for critical points in the closed interval -2, 2
Suppose we have a piecewise function
f(t) = exp(c*t) when 0 <= t < 2 and f(t) = 0 when t >= 2.
Can the above be rewritten as
f(t)= exp(at)*[H(t-0) - H(t-2)],
H is a heaviside function.
a) At which intervals are f strictly increasing and at what intervals are f strictly decreasing.
Should I just find the derivative of both of the functions? If so, I get that the function is increasing at the intervals (−∞,0) and (0,∞). Is this right, or can I just say that the function is...
Hello everyone.
I am trying to do a 2D Shannon interpolation, but I cannot use a sinc because later on this expression goes in an optimization software that doesn't recognize it. I have defined my own version of sinc as:
sincC = Piecewise[{(Sin[Pi* #]/(Pi*(#))), # >= 1}, {1 - (#^2)/6 +...
Problem Statement: Determine whether f is continuous at c.
(see image for piecewise function f)
EDIT: Sorry if it is a little blurry that is x^3 in the numerator of the rational function and x^2 in the denominator
Relevant Equations: Basic understanding of limits
My work:
Since the...
Homework Statement
Allow f:ℤ→ℤ be defined by, for all n∈ℤ
f(n) = {n-1 if n is even, n+5 if n is odd
Prove that ran(f) = ℤ
Homework EquationsThe Attempt at a Solution
I am unsure of how exactly to prove this due to the fact now I am working with a piecewise function.
Here is what I have so...
Homework Statement
Use the Heaviside function as an on the switch over the interval [a,b].
Homework Equations
Let the H(x) be the Heaviside function defined as a piece-wise function such that it is zero if x is less than zero, and 1 if it is greater than or equal zero. From that, we can use the...
Homework Statement
##f(x) = x \sin (\frac{1}{x})## for ##x \ne 0## and ##f(0) = 0##. Prove that this function is continuous at 0.
Homework EquationsThe Attempt at a Solution
First, I need to look at the quantity ##|f(x) - f(0)|##. However, I am not completely sure how to proceed. I would think...
Screenshot of my homework problem along with my solution so far. I'm not sure if I'm doing this correctly and if I am... if I'm answering correctly. Thank you. (EDIT: I made 1 small error with the piecewise definition. Ignore the f(x) before g(x).)
Hi PF!
I have a function ##\phi ns## defined below, and ##\phi ns## is continuous everywhere except ##s=0##, where it is singular. However, ##\lim_{s\to 0}\phi ns(s) = 0##, and since I'm integrating in this domain I thought I would define a new function ##\phi nsP## such that ##\phi...
Homework Statement
Hello I uploaded a picture of the image.
Find an expression for a function who's graph is the given curve.
Homework EquationsThe Attempt at a Solution
The first function [0,3] y= -x+3 and the second one is y=2x-6 from 3 to 5
what I don't get, is why isn't the second one...
Homework Statement
Find the values of a and b that make f a differentiable function.
Note: F(x) is a piecewise function
f(x):
Ax^2 - Bx, X ≤ 1
Alnx + B, X > 1
Homework EquationsThe Attempt at a Solution
Made the two equations equal each other.
Ax^2 - Bx = Alnx + B
Inserting x=1 gives,
A - B =...
Consider the function:
$$F(s) =\begin{cases} A \exp(-as) &\text{ if }0\le s\le s_c \text{ and}\\
B \exp(-bs) &\text{ if } s>s_c
\end{cases}$$
The parameter s_c is chosen such that the function is continuous on [0,Inf).
I'm trying to come up with a (unique, not piecewise) Maclaurin series...
Homework Statement
The function ##f## is defined as follows:
\begin{equation*}
f(t) =
\begin{cases}
1, \text{ when } 2k < t < (2k+1),\\
0, \text{ when } t = k,\\
2, \text{ when } (2k-1) < t < 2k, & k \in \mathbb{Z}\\
\end{cases}
\end{equation*}
What is the period ##T## of the function ##f##...
Homework Statement
a.) Let ##f,g:ℝ→ℝ## such that ##g(x)=sin x## and ##f(x)= \left\{
\begin{array}{ll}
x^2, x∈ℚ \\
0 , x∈ℝ\setminusℚ \\
\end{array}
\right. ##. Calculate ##\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}##.
b.) Why l'Hospital rule cannot be applied here?The Attempt at a...
Homework Statement
Write F(x)= x2-5|x| as a piecewise function
Homework EquationsThe Attempt at a Solution
I was writting it out and came to
Fx= x2-5(x) and x2-5(-x)
but my book says that it comes out to be
x2-5
x2-5(-x)
I imagine there is a very simple reason why the x in the first one...
Okay, so I'm down to the last equation.
-12/11 x 10 + 54/11 I get -66/-99. Is this right? If so how do I put it into the graph.
-12/11 x 10 = -120/110 + 54/11 = -66/99 (I think I've went wrong somewhere)
I recently plotted a piecewise function:
Plot[Piecewise[{{1 - Exp[-.002*t],
0 <= t < 120}, {-Exp[-.002*t] + Exp[-.002*(t - 120)],
120 <= t}}], {t, 0, 5000}, PlotRange -> {0, 0.25}]
I then defined the function which I am calling q[t_] as follows:
q[t_] := Piecewise[{{1 -...
Homework Statement
I've entered the following piecewise equation into Mathematica:
Plot[Piecewise[{{sin (t), 0 <= t < \[Pi]}, {5 + 5 cos (t) + sin (t), \[Pi] <= t < 4*\[Pi]}, {10 cos (t) + sin (t), 4*\[Pi] <= t}}], {t, 0, 20*\[Pi]}]
But I am getting a blank graph in return. I've proofread my...
Homework Statement
f(x)=-2 when x<1
=3 when x=1
=x-3 when x>1
find the limit at 1 from the left and right sides and at 1.
Homework EquationsThe Attempt at a Solution
limit for x when approaching 1 from the left is -2
limit for x when approaching 1 from the right is -2
-I'm not sure...
Homework Statement
f(t) = e^t when 0≤t<1
and 0 when t≥1
Homework Equations
Laplace transformations
The Attempt at a Solution
so the Laplace integral becomesfrom 0 to 1 ∫e^(st^2)dt + 0
how do I integrate this?
Write the piecewise function
\[ f(t) = \begin{cases}
2t, & 0\leq t < 3 \\
6, & 3 \le t < 5 \\
2t, & t \ge 5 \\
\end{cases}
\]
in terms of unit step functions.
So here is what i;ve got just guessing , I don't think I'm correct. I really need some help. But I got...
1. Problem
Define a function:
for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere}
for t<0 f(x,t) = - f(x,|t|)
Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x.
Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...
Dear Friends
I have a question about linear programming. It would be great to have your comments or suggestions.
Consider the followings
\begin{equation}
\\
Y = [y_{1}, y_{2}, \cdots, y_{n}]
\\
G = [g_{1}, g_{2}, \cdots, g_{n}]
\\
\textbf{X} =
\begin{pmatrix}
0 & x_{1,2} & \cdots & x_{1,n}...
Homework Statement
Homework Equations
The Attempt at a SolutionNot sure what to do here. I was thinking maybe the y – y1 = m(x – x1)? I am having trouble understanding this question. I know what piecewise functions are but filling this in is proving difficult.
Homework Statement
Given the piecewise function
f(x) = \left\{
\begin{array}{lr}
\frac{(2-x)^2-p}{x} &: x < q\\
r(x+6) &: q \leq x <2 \\
x^3-p &: x \geq 2
\end{array}
\right.
Find the values of p,q,r such that f(x) is continuous everywhere and f(2) = p
The Attempt at a Solution
Since f(2)...
The question looks like this.
Let $f(x, y)$ = 0 if $y\leq 0$ or $y\geq x^4$, and $f(x, y)$ = 1 if $0 < y < x^4 $.
(a) Show that $f$ is discontinuous at (0, 0)
(b) Show that $f$ is discontinuous on two entire curves.
In regarding (a), I know $f(x, y)$ is discontinuous on certain...
The question looks like this.
Let ##f(x, y)## = 0 if y\leq 0 or y\geq x^4, and f(x, y) = 1 if 0 < y < x^4 .
(a) show that f(x, y) \rightarrow 0 as (x, y) \rightarrow (0, 0) along any path through (0, 0) of the form y = mx^a with a < 4.
(b) Despite part (a), show that f is discontinuous at (0...
https://answers.yahoo.com/question/index?qid=20130915100124AAK4JAQ
I do not understand how they got:
"1 = |(1 plus d/2 - L) - (d/2 - L)| <= |1 plus d/2 - L| plus |d/2 - L| < 1/4 plus 1/4 = 1/2, "
Shouldn't it be $|(1+ \frac{\delta}{2} -L) + (\frac{\delta}{2} -L)|$, not $|(1+ \frac{\delta}{2}...
$$g(x)=\begin{cases}x^2, & \text{ if x is rational} \\[3pt] 0, & \text{ if x is irrational} \\ \end{cases}$$
a) Prove that $\lim_{{x}\to{0}}g(x)=0$
b) Prove also that $\lim_{{x}\to{1}}g(x) \text{ D.N.E}$
I've never seen a piecewise function defined that way...hints?
Homework Statement
Hey there. I'm trying to plot a piecewise function in MATLAB and I'm having some decent success, but there's some things I'm wondering.
Here's the function: http://gyazo.com/d3493a0c13096878acf5a501af8a7f66
Homework Equations
The Attempt at a Solution
My...
Hello MHB,
If I want to decide constant a and b so its continuous over the whole R for this piecewise function
basicly what I got problem with is that x^{1/3} is not continuous for negative value so it will never be continuous for any value on constant a,b. I am missing something? or do they...
Homework Statement
Prove the function ##f(x):[2,4]\rightarrow\mathbb{R}## defined by ##f(x) =\begin{cases} x, & \text{if }2\leq x\leq 3 \\2, & \text{if } 3<x\leq 4 \end{cases}## is Riemann IntegrableHomework Equations
A function ##f:[a,b]\rightarrow\mathbb{R}## is integrable iff for each...
Homework Statement
Make a piecewise function. If possible, please check my work.
- Salesperson has salary of $500.
- On first 10 000 of sales, earns 10% commission.
- On next 10 000 of sales, earns 20% commission.
- Earns 25% commission on any additional sales.
(note: didn't simplify to...
Homework Statement
Given that I have a doubling function :
f(x)=2x (for 0≤x<0.5) and 2x-1 (for 0.5≤ x<1)
Homework Equations
What is f(f(x))?
The Attempt at a Solution
f(f(x))=4x for the first one and 4x-3 for the second part but not sure what to do about the domain...
Homework Statement
Discuss the continuity and differentiability of
f(x) =
\begin{cases}
x^2 & \text{if } x\in \mathbb{Q} \\
x^4 & \text{if } x\in \mathbb{R}\setminus \mathbb{Q}
\end{cases}
Homework Equations
The Attempt at a Solution
From the graph of ##f##, I can see...