The space of continuous functions from r to r is a vector space under the natural operations of addition and scalar multiplication (you do not have to show this fact). determine wether the following subsets are subspaces of this space. justify your answers. (a) w1 = {f : r → r continuous , such that f(0) = f(1) and f(2) = f(3)}. (b) w2 = {f : r → r continuous , such that r 1 0 f(x)dx = 0}. (c) w3 = {f : r → r continuous , such that f(0) + f(2) = 1}.

Stu and ambrose are working together on a study of whether diet is related to students' mental health. first, they randomly select 10 students from the student center and 10 students from the office of counseling and psychological services. then, they ask students to rate their diet in terms of healthiness on a scale of 1-100 before completing a 20-item mental health survey. a) name one of the constructs being studied b) how was the construct you named above operationally defined in this experiment? e) was the variable you described above their independent variable, their dependent variable, or a controlled variable? d) what is a confounding variable? what is a potential confound in this study?