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 SolutionsQuestion 1True or False. The two functions f and g defined byare equal? Answer : False. Two functions are equal if their rules are equal and their domains are the same.
Question 2If 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 3True or False. The graph of f(x) and that of f(x + 2) are the sameAnswer : False. The graph of f(x + 2) is that of f(x) shifted 2 units to the left.
Question 4Let 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 5Let 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 add  4 to both sides on the inequality to obtain 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.Answer : True.
Question 7True 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.
