To find f(5) we substitute x=5 for x in f(x).

So we have to figure out which of these 3 intervals number 5 belongs to:

-7≤x<2, x=2, or x≥2 

We ask the question: "Does the number 5 fit the inequality -7≤x<2 ? 
The answer is NO, because 5 is not between -7 and 2, so that
means that we don't use f(x)=x²+3, because we only would use that one
if the value of x we wanted to substitute were between -7 and 2.  

We ask the question: "Does the number 5 fit the equation x=2?
No because 5 is not 2, so that means that we don't use f(x)=14, 
because we only would use f(x)=14 if the value of x we want 
to substitute were exactly 2.  

We ask the question: "Does the number 5 fit the inequality x≥2 ? 
The answer is YES, because 5 is INDEED greater than 2, so that
means that we DO use f(x)=4-3x² for that's the one we use if the value
of x that we want to substitute is greater than 2, and 5 surely is
greater than 2.

So we substitute 5 for x in f(x)=4-3x²  and we get

f(5)=4-3(5)² = 4-3(25) = 4-75 = -71.

Edwin