Question 1

Given sets $$A = \{1, 2\}$$ and $$B = \{a, b\}$$, which of the following correctly represents the Cartesian product $$A 	imes B$$?

Accepted Answer

$$\{(1, a), (1, b), (2, a), (2, b)\}$$

Question 2

Which of the following statements best describes a function $$f: A \rightarrow B$$?

Accepted Answer

For every $$x \in A$$, there is exactly one $$y \in B$$ such that $$(x, y) \in f$$

Question 3

If $$f(x) = x^2$ and x$$ \in \mathbb{R}$$, what is the image (range) of f$?

Accepted Answer

$$[0, \infty)$$

Question 4

Which of the following is NOT a requirement for a relation $$f to be a $$function from $$A to$$ $$B$$?

Accepted Answer

Each $$y \in B$$ must be associated with exactly one $$x \in A$$

Question 5

Consider the function $$f(x) = \sqrt{x}$$ with domain $$\mathbb{R}. $$What is the correct domain for $$f$$?

Accepted Answer

$$[0, \infty)$$

Question 6

A function $$f: A \rightarrow B is $$called injective (one-to-one) if:

Accepted Answer

If $$f(x_1$) = f(x_2$)$$, then $$x_1$$ = $$x_2$$

Question 7

Which of the following is an example of a surjective (onto) function $$f: \mathbb{R} \rightarrow \mathbb{R}$$?

Accepted Answer

$$f(x) = x$$

Question 8

If $$f: A \rightarrow B is $$bijective, which of the following statements is true?

Accepted Answer

Every $$y \in B is $$the image of at least one $$x \in A$$, and $$f is $$injective

Question 9

Given $$f(x) = x^2$ and g(x) = x + 1$, what is the composition f$$ \circ g (x)$$?

Accepted Answer

$$(x + 1)^2$

Question 10

A medical researcher models the growth of bacteria with $$f(t) = 2^t$$, where $$t is $$time in hours. What is the image of $$f$$ for $$t \in \mathbb{R}$$?

Accepted Answer

$$(0, \infty)$$

Question 11

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is the domain of the composition g$$ \circ f(x)$$?

Accepted Answer

$$\mathbb{R}$$

Question 12

A function $$f: \mathbb{R} \rightarrow \mathbb{R} is $$defined by $$f(x) = x^3$. Which of the following is true?

Accepted Answer

The function is bijective

Question 13

In a real-world scenario, a company models profit as $$P(x) = 5x - 10$$, where $$x is $$the number of products sold. What is the domain of $$P if $$only whole numbers of products can be sold?

Accepted Answer

$$\mathbb{N}$$

Question 14

Which of the following relations is NOT a function?

Accepted Answer

$$\{(1, 2), (1, 3), (2, 4)\}$$

Question 15

If $$f(x) = x^2$ and x$$ \in [-2, 2]$$, what is the image of f$?

Accepted Answer

$$[0, 4]$$

Question 16

A function $$f: A \rightarrow B is $$not surjective if:

Accepted Answer

Some $$y \in B$$ are not images of any $$x \in A$$

Question 17

In a chemical reaction, the concentration of a substance is modeled by $$C(t) = 10e$$^{-t}$$, t$$ \geq 0$$. What is the image of C$?

Accepted Answer

$$(0, 10]$$

Question 18

If $$f(x) = x^2$ and g(x) = x - 2$, what is f(g(3$$))$$?

Accepted Answer

$$1$$

Question 19

Which of the following best describes the codomain of a function $$f: A \rightarrow B$$?

Accepted Answer

The set $$B$$

Question 20

A function $$f: \mathbb{R} \rightarrow \mathbb{R} is $$defined by $$f(x) = |x|. $$Which of the following is true?

Accepted Answer

The image of $$f is$$ $$[0, \infty)$$

Study Notes: Relations and Functions (Calculus I, Aula 3)

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