the graph needed is on the bottom left.
this is visible from the circles on the graph.
if a circle at a certain point of the graph is blank (white), it means that that point is not part of the graph.
for #t^2#, where #0< t <3#,
the values #0# and #3# for #t# are not included, since it is a strong inequality.
this means that where #t = 0# and #t = 3#, the #xy#-coordinates will not be included. because of this, the points where these two values would be, if included, are blank circles instead.
the domain of the function is #0 < t <= 21.5#.
between these values, there are defined #y#-values for every value of #t# - there are no gaps between the definitions of the functions given.
this means that the inequality signs, after the smallest #t#-value and before the largest #t#-value, can be used to find the domain.
the range of the function is #0 <= y <= 33#.
this can be found from the largest and smallest #y#-values for the function #y = -2t + 43, 5 <= t <= 21.5#
#t = 5: y = 43 - 10 = 33#
#t >= 5: y <= 33#
#t = 21.5: y = 43 - 43 = 0#
#t <= 21.5: y >= 0#
#q(1) = 1^2 = 1#
#q(3) = 9 + 9 = 18#
#q(5) = 33#
#q(7) = 43 - 14 = 29#