piecewise

Conditionally defined expression or function

Syntax

pw = piecewise(cond1,val1,cond2,val2,...)

pw = piecewise(cond1,val1,cond2,val2,...,otherwiseVal)

Description

pw = piecewise(cond1,val1,cond2,val2,...) returns the piecewise expression or function pw whose value is val1 when condition cond1 is true, is val2 when cond2 is true, and so on. If no condition is true, the value of pw is NaN.

pw = piecewise(cond1,val1,cond2,val2,...,otherwiseVal) returns the piecewise expression or function pw that has the value otherwiseVal if no condition is true.

Examples

Define and Evaluate Piecewise Expression

Define the following piecewise expression by using piecewise.

$$\mathit{y}=\{\begin{array}{ll}-1& \mathit{x}<0\\ 1& \mathit{x}>0\end{array}$$

syms x
y = piecewise(x < 0,-1,x > 0,1)

y = 
$$\{\begin{array}{cl}-1& \text{ if }x<0\\ 1& \text{ if }0<x\end{array}$$

Evaluate y at -2, 0, and 2 by using subs to substitute for x. Because y is undefined at x = 0, the value is NaN.

subs(y,x,[-2 0 2])

ans = $$\left(\begin{array}{ccc}-1& \mathrm{NaN}& 1\end{array}\right)$$

Define Piecewise Function

Define the following function symbolically.

$$\mathit{y}\left(\mathit{x}\right)=\{\begin{array}{ll}-1& \mathit{x}<0\\ 1& \mathit{x}>0\end{array}$$

syms y(x)
y(x) = piecewise(x < 0,-1,x > 0,1)

y(x) = 
$$\{\begin{array}{cl}-1& \text{ if }x<0\\ 1& \text{ if }0<x\end{array}$$

Because y(x) is a symbolic function, you can directly evaluate it for values of x. Evaluate y(x) at -2, 0, and 2. Because y(x) is undefined at x = 0, the value is NaN. For details, see Create Symbolic Functions.

y([-2 0 2])

ans = $$\left(\begin{array}{ccc}-1& \mathrm{NaN}& 1\end{array}\right)$$

Set Value When No Condition Is True

Set the value of a piecewise function when no condition is true (called otherwise value) by specifying an additional input argument. If an additional argument is not specified, the default otherwise value of the function is NaN.

Define the piecewise function

$$y=\{\begin{array}{ll}-2& x<-2\\ 0& -2<x<0\\ 1& otherwise.\end{array}$$

syms y(x)
y(x) = piecewise(x < -2,-2,(-2 < x) & (x < 0),0,1)

y(x) = 
$$\{\begin{array}{cl}-2& \text{ if }x<-2\\ 0& \text{ if }x\in \left(-2,0\right)\\ 1& \mathrm{ otherwise}\end{array}$$

Evaluate y(x) between -3 and 1 by generating values of x using linspace. At -2 and 0, y(x) evaluates to 1 because the other conditions are not true.

xvalues = linspace(-3,1,5)

xvalues = 1×5

    -3    -2    -1     0     1

yvalues = y(xvalues)

yvalues = $$\left(\begin{array}{ccccc}-2& 1& 0& 1& 1\end{array}\right)$$

Plot Piecewise Expression

Plot the following piecewise expression by using fplot.

$$y=\{\begin{array}{cc}-2& x<-2\\ x& -2<x<2\\ 2& x>2\end{array}.$$

syms x
y = piecewise(x < -2,-2,-2 < x < 2,x,x > 2,2);
fplot(y)

Assumptions and Piecewise Expressions

On creation, a piecewise expression applies existing assumptions. Apply assumptions set after creating the piecewise expression by using simplify on the expression.

Assume x > 0. Then define a piecewise expression with the same condition x > 0. piecewise automatically applies the assumption to simplify the condition.

syms x
assume(x > 0)
pw = piecewise(x < 0,-1,x > 0,1)

pw = $$1$$

Clear the assumption on x for further computations.

assume(x,"clear")

Create a piecewise expression pw with the condition x > 0. Then set the assumption that x > 0. Apply the assumption to pw by using simplify.

pw = piecewise(x < 0,-1,x > 0,1);
assume(x > 0)
pw = simplify(pw)

pw = $$1$$

Clear the assumption on x for further computations.

assume(x,"clear")

Differentiate, Integrate, and Find Limits of Piecewise Expression

Differentiate, integrate, and find limits of a piecewise expression by using diff, int, and limit respectively.

Differentiate the following piecewise expression by using diff.

$$\mathit{y}=\{\begin{array}{ll}1/\mathit{x}& \mathit{x}<-1\\ \sin \left(\mathit{x}\right)/\mathit{x}& \mathit{x}\ge -1\end{array}$$

syms x
y = piecewise(x < -1,1/x,x >= -1,sin(x)/x);
diffy = diff(y,x)

diffy = 
$$\{\begin{array}{cl}-\frac{1}{x^2}& \text{ if }x<-1\\ \frac{\cos \left(x\right)}{x}-\frac{\sin \left(x\right)}{x^2}& \text{ if }-1<x\end{array}$$

Integrate y by using int.

inty = int(y,x)

inty = 
$$\{\begin{array}{cl}\log \left(x\right)& \text{ if }x<-1\\ \mathrm{sinint}\left(x\right)& \text{ if }-1\le x\end{array}$$

Find the limits of y at 0 by using limit.

limit(y,x,0)

ans = $$1$$

Find the right- and left-sided limits of y at -1. For details, see limit.

limit(y,x,-1,"right")

ans = $$\sin \left(1\right)$$

limit(y,x,-1,"left")

ans = $$-1$$

Elementary Operations on Piecewise Expressions

Add, subtract, divide, and multiply two piecewise expressions. The resulting piecewise expression is only defined where the initial piecewise expressions are defined.

syms x
pw1 = piecewise(x < -1,-1,x >= -1,1);
pw2 = piecewise(x < 0,-2,x >= 0,2);
add = pw1+pw2

add = 
$$\{\begin{array}{cl}-3& \text{ if }x<-1\\ -1& \text{ if }x\in \left[-1,0\right)\\ 3& \text{ if }0\le x\end{array}$$

sub = pw1-pw2

sub = 
$$\{\begin{array}{cl}1& \text{ if }x<-1\\ 3& \text{ if }x\in \left[-1,0\right)\\ -1& \text{ if }0\le x\end{array}$$

mul = pw1*pw2

mul = 
$$\{\begin{array}{cl}2& \text{ if }x<-1\\ -2& \text{ if }x\in \left[-1,0\right)\\ 2& \text{ if }0\le x\end{array}$$

div = pw1/pw2

div = 
$$\{\begin{array}{cl}\frac{1}{2}& \text{ if }x<-1\\ -\frac{1}{2}& \text{ if }x\in \left[-1,0\right)\\ \frac{1}{2}& \text{ if }0\le x\end{array}$$

Extract Expressions and Conditions from Piecewise

Extract conditions and values from a piecewise expression by using the children function.

syms x
pw = piecewise(x < -2,sin(x),(-2 < x) & (x < 2),x,2 < x,exp(-x))

pw = 
$$\{\begin{array}{cl}\sin \left(x\right)& \text{ if }x<-2\\ x& \text{ if }x\in \left(-2,2\right)\\ {\mathrm{e}}^{-x}& \text{ if }2<x\end{array}$$

c = children(pw);
pwExpr = [c{:,1}]

pwExpr = $$\left(\begin{array}{ccc}\sin \left(x\right)& x& {\mathrm{e}}^{-x}\end{array}\right)$$

pwCond = [c{:,2}]

pwCond = $$\left(\begin{array}{ccc}x<-2& x\in \left(-2,2\right)& 2<x\end{array}\right)$$

Modify or Extend Piecewise Expression

Modify a piecewise expression by replacing part of the expression using subs. Extend a piecewise expression by specifying the expression as the otherwise value of a new piecewise expression. This action combines the two piecewise expressions. piecewise does not check for overlapping or conflicting conditions. Instead, like an if-else ladder, piecewise returns the value for the first true condition.

Change the condition x < 2 in a piecewise expression to x < 0 by using subs.

syms x
pw = piecewise(x < 2,-1,x > 0,1);
pw = subs(pw,x < 2,x < 0)

pw = 
$$\{\begin{array}{cl}-1& \text{ if }x<0\\ 1& \text{ if }0<x\end{array}$$

Add the condition x > 5 with the value 1/x to pw by creating a new piecewise expression with pw as the otherwise value.

pw = piecewise(x > 5,1/x,pw)

pw = 
$$\{\begin{array}{cl}\frac{1}{x}& \text{ if }5<x\\ -1& \text{ if }x<0\\ 1& \text{ if }0<x\end{array}$$

Solve System of Equations with Piecewise Expressions

Since R2025a

Define a system of equations that includes piecewise expressions. The first equation defines the variable $\mathit{y}$ for the two conditions $0<\mathit{x}<2$ and $2<\mathit{x}<4$. The second equation defines $\mathit{x}\cdot \mathit{y}$ for the same two conditions.

syms x y
eq1 = y == piecewise((0 < x) & (x < 2),x^2 + x - 2,(2 < x) & (x < 4),3/2*x)

eq1 = 
$$\{\begin{array}{cl}y=x^2+x-2& \text{ if }x\in \left(0,2\right)\\ y=\frac{3\hspace{0.17em}x}{2}& \text{ if }x\in \left(2,4\right)\end{array}$$

eq2 = y*x == piecewise((0 < x) & (x < 2),x - 1,(2 < x) & (x < 4),9)

eq2 = 
$$\{\begin{array}{cl}x\hspace{0.17em}y=x-1& \text{ if }x\in \left(0,2\right)\\ x\hspace{0.17em}y=9& \text{ if }x\in \left(2,4\right)\end{array}$$

To solve this system of equations, first substitute the variable $\mathit{y}$ in the second equation with the expression for $\mathit{y}$ from the first equation. You can use the lhs and rhs functions to get the left and right sides of the first equation.

eq2 = subs(eq2,lhs(eq1),rhs(eq1))

eq2 = 
$$\{\begin{array}{cl}x\hspace{0.17em}\left(x^2+x-2\right)=x-1& \text{ if }x\in \left(0,2\right)\\ \frac{3\hspace{0.17em}{x}^2}{2}=9& \text{ if }x\in \left(2,4\right)\end{array}$$

Then, solve the second equation for the variable $\mathit{x}$.

sols = solve(eq2,x)

sols = 
$$\left(\begin{array}{c}1\\ \sqrt{6}\\ \sqrt{2}-1\end{array}\right)$$

Here, $\mathit{x}=1$ and $\mathit{x}=\sqrt{2}-1$ are solutions that satisfy the condition $0<\mathit{x}<2$, while $\mathit{x}=\sqrt{6}$ is a solution that satisfies the condition $2<\mathit{x}<4$.

Input Arguments

cond — Condition

symbolic condition | symbolic variable

Condition, specified as a symbolic condition or variable. A symbolic variable represents an unknown condition.

Example: x > 2

val — Value when condition is satisfied

number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Value when condition is satisfied, specified as a number, vector, matrix, or multidimensional array, or as a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.

otherwiseVal — Value if no conditions are true

number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Value if no conditions are true, specified as a number, vector, matrix, or multidimensional array, or as a symbolic number, variable, vector, matrix, multidimensional array, function, or expression. If otherwiseVal is not specified, its value is NaN.

Output Arguments

pw — Piecewise expression or function

symbolic expression | symbolic function

Piecewise expression or function, returned as a symbolic expression or function. The value of pw is the value val of the first condition cond that is true. To find the value of pw, use subs to substitute for variables in pw.

Tips

• piecewise does not check for overlapping or conflicting conditions. A piecewise expression returns the value of the first true condition and disregards any following overlapping true expressions. Thus, piecewise follows the behavior of an if-else statement.

• In R2025a: You can use the lhs and rhs functions to get the left and right sides of a symbolic equation that includes a piecewise expression.

Version History

Introduced in R2016b