Now, I know that the function can also be written as: s/(s-0.5)

So I tried plotting the step and impulse responses in Matlab:

sys = tf([1 0],[1 -0.5])

figure(1);

step(sys);

figure(2);

impulse(sys);

However, both graphs look the same (can't post images of my graphs, I need more rep to do it).

Both graphs have exponential growth, but shouldn't the impulse response look like exponential decay?