Why is this function BIBO stable? I have a continuous system described as:
$$y(t)=2x(t)+0.5x(t-2)+0.25x(t+2)$$
and I'm trying to understand why the system is stable. I know that a system is unstable when you provide an input and as a result the output is not bounded. So, when I looked at the system I set $t=\infty$ and assumed that the output goes to infinity because I'd have
$$ y(\infty)=2x(\infty)+0.5x(\infty-2)+0.25x(\infty+2)$$
making it unbounded. Looking at this equation, isn't the output going to infinity resulting in an unstable system? The solutions state it's stable and I can't see why. Thanks
Triangle inequality is your friend: if $y = a_1x_1+  a_2x_2+ a_3x_3 $, then $$|y| \le |a_1||x_1|+  |a_2||x_2|+ |a_3||x_3|$$
Hence if $x_{.}$ is bounded, the sum is bounded as well.