# 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.

 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.

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Updated: February 2015