For the following question, given a collision resistant function $f$:$\{0,1{\}}^{1000}\{0,1{\}}^{256}.$

Define function $g$:$\{0,1{\}}^{1000}\{0,1{\}}^{128}$ as follows:

for every $x,g(x)$ first run $f(x),$ cut it into two strings, and throw away the last half, i$.$e$.g(x)$ outputs the first half of $f(x).$

Would $g$ be collision resistant? Please explain your answer. $($If your answer is "yes", please justify your reasoning; otherwise, please give a counterexample.