Conditionally defined expression or function (2024)

Define the following piecewise expression by using piecewise.

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

syms xy = 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 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 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

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

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 the following piecewise expression by using fplot.

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

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

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 xassume(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 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 xy = 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 =$$-1$$

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

syms xpw1 = 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 =
$$\{\begin{array}{cl}\frac{1}{2}& \text{if}x<-1\\ -\frac{1}{2}& \text{if}x\in \left[-1,0\right)\\ \frac{1}{2}& \text{if}0\le x\end{array}$$

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

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.