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.