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      FUNCTIONS

      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)

      (D) 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)