Question 1:
True or False. The two functions f and g defined by
f(x) = 3x + 3 for x real and g(t) = 3t + 3 for t real and positive
are equal?
Answer :
False. Two functions are equal if their rules are equal and their domains are the same.
Question 2:
If functions f and g have domains Df and Dg respectively, then the domain of f / g is given by
(A) the union of Df and Dg
(B) the intersection of Df and Dg
(C) the intersection of Df and Dg without the zeros of function g
(D) None of the above
Answer :
(C). Division by zero is not allowed in mathematics. Students tend to forget this point.
Question 3:
True or False. The graph of f(x) and that of f(x + 2) are the same
Answer :
False. The graph of f(x + 2) is that of f(x) shifted 2 units to the left.
Question 4:
Let the closed interval [a , b] be the domain of function f. The domain of f(x  3) is given by
(A) the open interval (a , b)
(B) the closed interval [a , b]
(C) the closed interval [a  3 , b  3]
(D) the closed interval [a + 3 , b + 3]
Answer :
(D). The graph of f(x  3) is that of f(x) shifted 3 units to the right. To shift the closed interval [a , b] to the right you need to add 3 units to the endpoints a and b of the interval.
Question 5:
Let the interval (a , +infinity) be the range of function f. The range of f(x)  4 is given by
(A) the interval (a  4 , +infinity)
(B) the interval (a + 4, +infinity)
(C) the interval (a, +infinity)
(C) None of the above
Answer :
(A). If the range of f is given by the interval (a , +infinity), we can write the following inequality
f(x) > a
add  4 to both sides on the inequality to obtain
f(x)  4 > a  4
The last inequality suggests that the range of f(x)  4 is (a  4, +infinity)
Question 6:
True or False. The equation y =  x  , with y >= 0, represents y as a function of x.
Answer :
True.
Question 7:
True or False. The equation x =  y  , with x >= 0, represents y as a function of x.
Answer :
False. Solve for y to find that y =  x  or y =  x ; for one value of the independent variable x we have two values of the dependent variable y.
More references on calculus
questions with answers, tutorials and problems and Questions on Functions with Solutions.
