# 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?
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
(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
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 $$[a + 3 , b + 3]$$
(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
(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 $$x$$.
True.

### Question 7

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