A set of questions on the concepts of a function, in calculus, are presented along with their solutions. These questions have been designed to help you gain deep understanding of the concept of a function as well as its domain and range which are of major importance in mathematics. If you have difficulties answering a question, review the definition(s) and theorem(s) related to the question.

Questions with Solutions

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.