Define and Evaluate Piecewise Expression

Define the following piecewise expression by using piecewise.

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

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

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 =
[ -1, NaN, 1]

Define Piecewise Function

Define the following function symbolically.

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

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

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 =
[ -1, NaN, 1]

Set Value When No Conditions 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

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

y(x) =
piecewise(x < -2, -2, x in Dom::Interval(-2, 0), 0, 1)

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)
yvalues = y(xvalues)

xvalues =
    -3    -2    -1     0     1
yvalues =
[ -2, 1, 0, 1, 1]

Plot Piecewise Expression

Plot the following piecewise expression by using fplot.

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.

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

diffy =
piecewise(x < -1, -1/x^2, -1 < x, cos(x)/x - sin(x)/x^2)

Integrate y by using int.

inty = int(y, x)

inty =
piecewise(x < -1, log(x), -1 <= x, sinint(x))

Find the limits of y at 0 and -1 by using limit. Because limit finds the double-sided limit, the piecewise expression must be defined from both sides. Alternatively, you can find the right- or left-sided limit. For details, see limit.

limit(y, x, 0)
limit(y, x, -1)

ans =
1
ans =
limit(piecewise(x < -1, 1/x, -1 < x, sin(x)/x), x, -1)

Because the two conditions meet at -1, the limits from both sides differ and limit cannot find a double-sided limit.

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
sub = pw1 - pw2
mul = pw1 * pw2
div = pw1 / pw2

add =
piecewise(x < -1, -3, x in Dom::Interval([-1], 0), -1, 0 <= x, 3)
sub =
piecewise(x < -1, 1, x in Dom::Interval([-1], 0), 3, 0 <= x, -1)
mul =
piecewise(x < -1, 2, x in Dom::Interval([-1], 0), -2, 0 <= x, 2)
div =
piecewise(x < -1, 1/2, x in Dom::Interval([-1], 0), -1/2, 0 <= x, 1/2)

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 =
piecewise(x < 0, -1, 0 < x, 1)

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 =
piecewise(5 < x, 1/x, x < 0, -1, 0 < x, 1)