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.

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  1. A

    Converting Piecewise function to Heaviside equation

    Good day! I would just like to ask on how to convert a particular piece wise function to a heaviside equation. For a function, f(t) = t2 if t < 2 t - 1 if t >= 2 the heaviside equation would be, H(t) = t + (t-1 -t)u2(t) or H(t) = t - u2(t) but i can't comprehend on...
  2. K

    Evaluation of Limit of Piecewise function help needed

    Homework Statement Find the continuity of a piece wise function f(x) = (x^2-9)/(x-3) for x not equal to 3 and f(x) = 6 for x=3...the continuity is to be found at x = 3 Homework Equations Limit f(x) as x--> c = LThe Attempt at a Solution By seeing the first rule it is clearly...
  3. D

    Piecewise Function Homework: Solve Fourier Coefficient b1

    Homework Statement Given the following periodic signal: What is the Fourier coefficient b1. I'm not asking for help with the Fourier series. I feel my integration is solid. What I need help with is creating the correct function! The Attempt at a Solution I first declare t=5 as now to be...
  4. paulmdrdo1

    MHB Solving Piecewise Functions with U(x) - U(x-1): Explanation and Graph

    given function U(x) U(x) = { 0, if x<0 {1, if x>=0 define the fuction piecewise and graph. * U(x) - U(x-1) how to solve this? please explain! thanks!
  5. TheBigBadBen

    LaTeX How can I write a piecewise function in LaTeX?

    I was trying to type a piecewise function, and then this happened: f(x)=\left{ \begin{array}{ccc} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q} \end{array} \right What's going on, and how do I fix it?
  6. D

    Fourier Series for a piecewise function help

    Homework Statement I'm trying to find a Fourier series for the piecewise function where f(x)= 0 \in -\pi \leq x \leq 0 -1 \in 0 \leq x \leq \frac{\pi}{2} 1 \in \frac{\pi}{2} \leq x \leq \pi Homework Equations a_{n} = \frac{1}{\pi} \int_{0}^{2\pi}\cos(nx)y(x)\,dx b_{n} = \frac{1}{\pi}...
  7. Z

    Find values for this piecewise function so it can be differentiable

    Homework Statement Fx= ax^2 + bx + c -infiniti<x<0 = D x=0 =x^2 sin(1/x) - 2 0<x<infiniti a) FInd all vlaues of a, b, c and d that make the function f differntiable on the domain -∞<x<∞ b) Using the values founbd in part a, determine lim x->...
  8. fluidistic

    Laplace transform of a piecewise function

    Homework Statement I must calculate the Laplace transform of the following function: f(x)=1 for x \in [0,1] \cap [2,3] \cap [4,5] \cap ... , f(x)=0 otherwise. Homework Equations The Laplace transform is \mathbb{L} [f(x)]=\int _0 ^{\infty} e^{-sx}f(x)dx. The Attempt at a Solution...
  9. D

    Piecewise function - create function from graph

    Homework Statement Hope I named the problem accurately. I'm trying to calculate the average value, and the RMS value of this voltage. The Attempt at a Solution I've been given the graph, and I know how to calculate average and RMS. That's not the problem. What I am wondering...
  10. J

    Fourier Series of an Odd Piecewise function

    Homework Statement Fourier Series of the following function f(x). f(x) is -1 for -.5<x<0 f(x) is 1 for 0<x<.5Homework Equations b_{n} = \frac{1}{L}\int^{L}_{-L}f(x)sin(nπx/L)dx Where L is half the period.The Attempt at a Solution Graphing the solution, I know that it is odd, which is why I...
  11. F

    Piecewise function radians and numbers ?

    Hello, I have encountered the following piecewise function: (x − 1)^2 if x > 1 tan x if −π/2 < x ≤ 0 How would I interpret/graph this, as it seems to be referring to radians and regular numbers ? Thanks for any help.
  12. J

    Mathematica [Mathematica] NIntegrate Piecewise function with DE solution

    Hi, I'd like to solve a DE, create a function with the solution, then use that solution in a Piecewise function, and then NIntegrate the Piecewise function but I can't get NIntegate to work. Here's what I'm trying to do: mysol = NDSolve[{y'[x] == x, y[0] == 1}, y, {x, 0, 1}] myy[x_] :=...
  13. E

    Graphing a piecewise function with multiple functions

    Homework Statement Suppose f(x) is a piecewise function defined as follows f(x) = 2x^2+2 ---- > x < 1 = 2x^2 - 3x ----- > 1 ≤ x ≤ 2 = 2 - (6/x) ----- > x > 2 Graph f(x) for 0 ≤ x ≤ 3Find the left and right hand limits of f(x) as x approaches 1 and as x approaches...
  14. I

    Where is this piecewise function discontinuous?

    g(x) = 3 if x < -4 g(x) = 7+x if |x| <or= 4 g(x) = x^4 if x > 4 I know there is a jump discontinuity at x=4. How would I state that in interval notation? Is that even possible or is it good enough to say g(x) is discontinuous at x=4? Basically what I've found is that if x=4 then g(x)=11...
  15. T

    Writing an absolute value function as a piecewise function

    Homework Statement Write f(x) = |x2-x-12| as a piecewise function. Homework Equations The Attempt at a Solution -x2+x+12 where x>1/2 x2-x-12 where x\geq0 According to the answer book the answers are -x2+x+12 where -3<x<4 x2-x-12 where x\geq4 2x2-5x-3, x\leq1/2 I am...
  16. G

    Limits and Continuity of a Piecewise Function

    Homework Statement Find a value for k to make f(x) continuous at 5 f(x)= sqrt(x2-16)-3/(x-5) if x cannot equal 5 3x+k when x=5 Homework Equations none The Attempt at a Solution lim x->5 sqrt((x+4)(x-4))-3/(x-5) * sqrt((x+4)(x-4))+3/sqrt((x+4)(x-4))+3 lim x->5...
  17. B

    Piecewise function question

    Homework Statement With reference to this graph: http://imageshack.us/photo/my-images/41/deletemev.jpg/ Is the value of c that makes lim x->1 [f(x) + c g(x)] exist equal to the same value of c that makes f(x) + c g(x) continuous at x = 1? Homework Equations The Attempt at a...
  18. I

    Derivative of this piecewise function?

    Homework Statement I have this piecewise defined function and am wondering if the derivative is supposed to be piecewise defined as well? Or could I just write it out on a line like a "normal" function? Homework Equations x(t) = {-14t : 0\leq|t|< 1, 2sgn(t) : 1\leq|t|< 7, (-12/7)sgn(t) ...
  19. D

    Curve fitting piecewise function

    I have a piecewise function described by g_1(x) as shown in the figure below. I wish to make a smooth curve, g_2(x), to fit (but not necessarily exactly) g_1(x). The only conditions are: Both curves must start at (0,0) and end at (a,0). The area of both curves must be the same. How do...
  20. J

    Find the Values of a and b in a piecewise function. Help please

    Find the values of a and b that make the function f(x) differentiable. f(x)={ ax^3 + 1 , for x<2 { b(x-3)^2 +10, for x is greater than or equal to 2 ax^3 + 1 = b(x-3)^2 +10 3ax^2 = 2b(x-3) From here I'm unsure of where to go.
  21. I

    Fourier series of a piecewise function

    i'm going through this question. firstly how would the graph look like? at first i thought it would be http://i.imgur.com/XfVCK.png the top graph drawn, but then i thought maybe it's the bottom one. if it was the bottom one then it'd be easier for me to do the question, since it'd be an odd...
  22. W

    Continuity on piecewise function

    Homework Statement [10 Marks] At which points is the following function continuous and at which point is it discontinuous. Explain the types of discontinuity at each point where the function is discontinuous. Then at each point of the discontinuity, if possible, find a value for f(x) that makes...
  23. J

    How do I write this interesting piecewise function?

    Homework Statement http://www2.seminolestate.edu/lvosbury/images/VibSpringAnNS.gif Find the governing differential equation and position functions for a 32 pound object attached to the end of a spring with a spring constant of 1 and a forcing function that yields a constant velocity in the...
  24. R

    Piecewise Function Problem Clarification.

    Homework Statement A hotel chain charges $95 each night for the first two nights and $55 for each additional nights stay. The total cost T is a function of the number of nights x that a guest stays. (a) Find the expressions for a and b in the piecewise function (b) Find T(2), T(3), and T(5)...
  25. E

    Sigma algebra generated by a piecewise function

    I have this homework question and I'm a little bit stuck. The question is: Let (Ω, F , P ) be the unit interval probability space, and X be the random variable defined by (See attatchment) Find the σ-algebra generated by X. From my understanding of sigma-algebras I want to look...
  26. B

    How to square a piecewise function? :/

    how to square a piecewise function? urgent :/ Homework Statement square the following piecewise function f(x)= 0 [-pi,-pi/2) pi [-pi/2,pi/2) 0 [pi/2,pi) Homework Equations N/A The Attempt at a Solution so i think this is how you do it: (0)2 + (pi)2 +(0)2 is...
  27. J

    What is the derivative of ln(e^(x^2 + 2))?

    Hey everyone, alright so i have two calculus questions that i need help with: 1. the Integral (lower bound 'a' , upper bound 'b') of f(x) dx = a + 2b Then the Integral (lower bound 'a' , upper bound 'b') of (f(x) + 3) dx = ? I have no idea what the second integral would equal...
  28. D

    Making Piecewise Function Continuous w/ First & Second Derivatives

    I have to make a piecewise function continuous with first and second derivatives at x=0 by finding a value for a and b. I have made the function continuous by equating the RHS and LHS limits and then solving for the variable a. The limit of the first derivatives of the LHS and RHS are the...
  29. Repetit

    Continuity of piecewise function undefined for 1<x<=2

    My math book claims that the piecewise function f : [0,1] U (2,3] --> R defined by f(x)= x for 0<=x<=1 x-1 for 2<x<=3 is continuous. But it's undefined for 1<x<=2 so how can it be continuous? According to the definition of continuity, a function is at a point x0 if for a sequence x_n...
  30. W

    Piecewise Function Integration

    Homework Statement Find formulas for the upper and lower sums of f on P_n, and use them to compute the value of \int_0^1f(x)dx. P_n:=\{\frac{j}{n}:j=0,1,...,n\} (a partition of [0,1]) \[ f(x) = \left\{ \begin{array}{ccc} 0 & 0 \le x < 1/2 \\ 1 & 1/2 \le x \le 1 \end{array} \right. \]Homework...
  31. J

    Construct a piecewise function f(x)

    Homework Statement Construct a piecewise function f(x) such that the following conditions are satisfied: Concave down on ( - infinity, 5) Concave up on (5, infinity) increasing over ( -infinity,infinity) Nonzero- sloped tangent line at the point of inflection x = 5 f(5) = 11 (continuous at x =...
  32. T

    MATLAB Defining piecewise function in matlab

    How would you write piecewise functions in MATLAB that can take vector inputs? Here's a function that I'm trying to write. function y=g(x) if x==0 y=1; else y=sin(x)./x; end If I call g([0,pi/2]), I want it to return [0,2/pi], but what I get instead is [NaN,2/pi]. I'm guessing...
  33. O

    Solving for A and B in a piecewise function

    Homework Statement ax+b, x>-1 f(x)= bx^2-3, x less than equal to: -1 Homework Equations the limits on both sides The Attempt at a Solution found the limit on both sides of the equation but don't know what to do next.
  34. G

    Differentiation of a Piecewise Function

    Homework Statement I have a graph of velocity vs time and was asked "Describe how the velocity changes over the time interval of the problem" how do I do this all I have is the graph that is not defined by a function... well I can make my own strangely all it is is a bunch of linear lines...
  35. J

    Continuity of piecewise function f(x,y)

    Homework Statement Determine all points at which the given function is continuous. For practice, I want to verify the continuity. Moreover, with piecewise function, I have to verify continuity anyway. Q1 Q2 Homework Equations The Attempt at a Solution Let's do the second problem first...
  36. F

    Piecewise function of the following function given

    Homework Statement Question: Rewrite the function f(x) =||x|-1| as a piecewise function. Homework Equations The Attempt at a Solution I tried considering the following absolute value functions: 1)|x| |x| = x, if x \geq 0 = -x, if x < 0 2) |x-1| |x-1| = x-1, if x \geq...
  37. M

    Making a piecewise function continuous

    Homework Statement find the values of b and c that make the function f continuous on (-\infty,\infty) f(x) = \frac{sin2x}{x} if x< 0 3-3c+b(x+1) if 0\leqx<2 5-cx+bx^2 if x\geq 2 Homework Equations lim as x...
  38. S

    Inverse of a Piecewise Function

    Homework Statement Find the inverse of ...{x...x =/= a1,...,an f(x) = {ai+1...x = ai, i = 1,...,n-1 ...{a1...x = an Homework Equations The Attempt at a Solution I interchanged the variables x and y, but I am very confused as to how to solve for the rest. I don't understand how to find...
  39. R

    Diffrentiating Piecewise Function

    Homework Statement Let L\in R and define h(x) = \begin{cases} sin(1/x) & \text{ if } x \neq 0 \\ L & \text{ if } x = 0 \end{cases} Prove that h is not continuous at 0 Homework Equations My Professor gave us the hint "Prove by contradiction, taking \epsilon...
  40. S

    Piecewise function - Find derivative at 3

    In this question, we shall take steps to find the values of a and b , given that the function f(x)={ x^2−4x+1 if x<=3 ax+b if x>3 is differentiable at 3. a) It is known that if a function is differentiable at a point c, then it is continuous at c. Using now the continuity of f at 3...
  41. D

    Double integral of piecewise function

    Homework Statement Let f(x,y)= 1 if x is rational, 2*y if x is irrational Compute both double integrals of f(x,y) over [0,1]x[0,1] Homework Equations The Attempt at a Solution I'm tempted to say that we can do the dydx integral since when x is rational, integrating y gives...
  42. P

    1D Heat Equation BC: Both ends insulated, IC: piecewise function

    1. Solve the one dimensional heat equation for a rod of length 1 with the following boundary and initial conditions: BC: \partialu(0,t)/\partialt = 0 \partialu(1,t)/\partialt = 0 these are the wrong boundary conditions (see below) Actual BC: \partialu(0,t)/\partialx = 0...
  43. T

    Partial Derivative, piecewise function

    Homework Statement Let : f(x,y) = \frac{xy(x^2 - y^2)}{(x^2 + y^2)^2} if (x,y) \neq (0,0) f(x,y) = 0 if (x,y) = (0,0) a) Find f_{xx}(0,0) b) Find f_{xy}(0,0) c) Find f_{yx}(0,0) Homework Equations None The Attempt at a Solution I'm not sure how to deal with the piecewise...
  44. N

    Describing a piecewise function

    Homework Statement Hi all I have a function f(x), which is: -1 when x is in the intervals .. (-4;-2), (0;2), (4;6) etc.. and 1 when x is in the intervals ... (-6;-4), (-2;0), (2;4) etc... At the points -4,-2, 0, 2 .. there are discontinuities. 1) How would I write this function...
  45. U

    How to Calculate Engraving Costs for Pewter Mugs: An Engraving Promotion Guide

    We need help on piecewise function... A gift shop sells pewter mugs for $35. They are currenly running an engraving promotion. The first six letter are free. Each additional letter cost $0.20. Write the piecewise model that gives the price of the mug with x engraved letter.. PLease help we...
  46. M

    Help me to verify the piecewise function answers

    I am trying to solve the exercise in this link, please let me know if i am doing something wrong. ( i didn't find answers to this exercise that's why i am sending it here) http://www.libraryofmath.com/limits-with-piecewise-functions.html exercise 1 ---------- a) lim x->3- f(x) = 7 b)...
  47. Hootenanny

    Maple Maple plot piecewise function for a mark scheme

    I need to plot the following piecewise function for a mark scheme I'm writing, but I can't seem to get maple to plot it. I can define the function, but when I ask maple to plot it, it just hangs. Any help would be much appreciated. The function; f(\theta):= \left\{\begin{array}{lcr}...
  48. D

    Absolute Value functions => Piecewise Function

    Given the function: f(x) = \left| {x + 1} \right| + \left| {x + 2} \right| How can i write that as a piecewise function? If i was given something in the form of f(x) = \left| {g(x)} \right|, i know to write it as: f(x) = \left\{ {\begin{array}{*{20}c} {g(x),} & {g(x) \ge 0}...
  49. T

    Help with Piecewise function, i had a bronchitis

    Homework Statement I had a bronchitis the past week, and I have no idea what to do with this problem, my teacher gave us two problems out of 28 from the upcoming quiz, and this is one of the two. What the following absolute value using a piecewise function? y=|x+5|+2 Express the area...
  50. P

    Continuously Differentiable Piecewise Function?

    Here is a piecewise polynomial function: f(x) = x^2 + 1 if x <= 1 f(x) = 2x if x > 1 I need to prove that this function is differentiable at x = 1? It's a parabola that turns into a line. It doesn't have any gaps or corners. The limit of f(x) as x approaches 1 is 2, and the limit of...
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