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 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 \( D_f \) and \( D_g \) respectively, then the domain of \( f / g \) is given by
(A) the union of \( D_f \) and \( D_g \)
(B) the intersection of \( D_f \) and \( D_g \)
(C) the intersection of \( D_f \) and \( D_g \) excluding 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 \( (a , + \infty)\) be the range of function \( f \). The range of \( f(x) - 4 \) is given by
(A) the interval \( (a - 4 , + \infty ) \)
(B) the interval \( (a + 4, + \infty) \)
(C) the interval \( (a, + \infty) \)
(D) None of the above
Answer :
(A).
If the range of \( f \) is given by the interval \( (a , +\infty) \), 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 \( (a - 4, + \infty) \)

Question 6

True or False . The equation \( y = | x | \), with \( y \geq 0 \), represents \( y \) as a function of \( x \).
Answer :
True.

Question 7

True or False . The equation \( x = | y | \), with \( x \geq 0 \), 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 \( y \) and therefore \( x = | y | \) does not represents \( y \) as a function of \( x \).

References and Links

Calculus questions with answers ,
Calculus tutorials and problems
Questions on Functions with Solutions .

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