Questions and Answers on Functions
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 1True 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
False. Two functions are equal if their rules are equal and their domains are the same.
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
(C). Division by zero is not allowed in mathematics. Students tend to forget this point.
Question 3True or False. The graph of f(x) and that of f(x + 2) are the same
False. The graph of f(x + 2) is that of f(x) shifted 2 units to the left.
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]
(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.
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
(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 6True or False. The equation y = | x | , with y >= 0, represents y as a function of x.
True or False. The equation x = | y | , with x >= 0, represents y as a function of x.
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.