syms x(t) shifted_x(t)
x(t) = evalin(symengine, 'piecewise([0 <= t and t < 2, exp(t)/2], [2 <= t and t < 4, 2*t-8], [t < 0 or t >= 4, 0])');
shifted_x(t) = x(-t-1);
I do not recommend this approach. The evalin(symengine) is ugly and relies upon a feature that is probably going to be removed. It is, however, the only way to create a symbolic piecewise . If you need to do something like integrate this or laplace transform then you would need the symbolic version.
If you do not need symbolic manipulation then you should use logical indexing, possibly together with a "trick":
x = @(t) (0 <= t & t < 2) .* exp(t)/2 + (2 <= t & t < 4) .* (2*t-8);
shifted_x = @(t) x(-t-1);
This trick will not work if any of your formula can produce a NaN or inf at a location where the piecewise is not intended to apply. For example,
(x ~= 0) .* 1./x + (x == 0) .* (-1) .* ones(size(x))
will fail if x becomes 0, because even though the (x ~= 0) seems to mask out the 1./x, the 1./x will be evaluated anyhow and will return inf and the false of (x ~= 0) multiplied by inf will give NaN.
But for example,
(x > 0) .* 1./max(x, eps) + (x <= 0) .* (-1) .* ones(size(x))
would be okay because at those negative and 0 values, max(x,eps) would be a non-zero value and 1 divided by that value would not be inf or nan, and then the (1/eps) that resulted would be multipled by the false of (x > 0) giving 0 which is fine.
