Q: Suppose $$f(x) = \frac{1}{x^2$ - 4}. $$What is the domain of $$f$$?

$$x \in \mathbb{R}, x \neq 2, x \neq -2$$

Q: Which of the following is the correct set notation for the set of all $$x$$ such that $$x is a $$rational number?

$$Q = \{x \mid x \text{ is rational}\}$$

Q: Which of the following is the correct set notation for the intersection of sets $$A$$ and $$B$$?

$$A \cap B = \{x \mid x \text{ belongs to both } A \text{ and } B\}$$

Question 1

Which of the following sets contains all real numbers $$x$$ such that $$-1 \u003c x \u003c 6$$?

Accepted Answer

$$B = \{x \mid -1 \u003c x \u003c 6\}$$

Question 2

What is the domain of the function $$f(x) = \sqrt{x-2}$$?

Accepted Answer

$$x \geq 2$$

Question 3

Suppose $$f(x) = \frac{1}{x^2$ - 4}. $$What is the domain of $$f$$?

Accepted Answer

$$x \in \mathbb{R}, x 
eq 2, x 
eq -2$$

Question 4

Which of the following is the correct definition of the distance between two real numbers $$a$$ and $$b$$?

Accepted Answer

$$|a - b|$$

Question 5

A function $$f is $$defined by $$f(x) = mx + b. $$Which statement is true about its graph?

Accepted Answer

It is a straight line with slope $$m$$ and $$y-$$intercept $$b.$$

Question 6

Given the function $$f(x) = \frac{1}{x}$$, which of the following is true about its domain?

Accepted Answer

$$x \in \mathbb{R}, x 
eq 0$$

Question 7

Which of the following functions passes the vertical line test but fails the horizontal line test?

Accepted Answer

$$y = x^2$

Question 8

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

Accepted Answer

$$x \in \mathbb{R}$$

Question 9

Which of the following is NOT a rational number?

Accepted Answer

$$\sqrt{2}$$

Question 10

A real-world application: The distance $$d$$ between two cars on a straight road is given by $$d = |x_1$ - x_2$|$$, where $$x_1$$ and $$x_2$$ are their positions. If car A is at $$x_1 = 120$ meters and car B is at x_2$ = 85$$ meters, what is their distance?

Accepted Answer

$$35$$ meters

Question 11

Which of the following intervals includes all real numbers $$x$$ such that $$2 \leq x \u003c 5$$?

Accepted Answer

$$[2, 5)$$

Question 12

If $$f(x) = x^2$ - 4x + 3$$, what is the range of $$f$$?

Accepted Answer

$$f(x) \geq -1$$

Question 13

Which of the following is the correct set notation for the set of all $$x$$ such that $$x is a $$rational number?

Accepted Answer

$$Q = \{x \mid x 	ext{ is rational}\}$$

Question 14

Which of the following is the correct formula for the slope $$m of a $$line passing through points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

$$m = \frac{y_2$ - y_1$}{x_2$ - x_1$}$$

Question 15

Which of the following is the correct definition of the range of a function $$f$$?

Accepted Answer

The set of all possible values $$f(x)$$ can take.

Question 16

A function $$f is $$defined by $$f(x) = \frac{1}{x-3}. $$What is the domain of $$f$$?

Accepted Answer

$$x \in \mathbb{R}, x 
eq 3$$

Question 17

Which of the following is the correct interval notation for all real numbers $$x$$ such that $$x \u003e 0$$?

Accepted Answer

$$(0, \infty)$$

Question 18

If $$f(x) = x^2$ and g(x) = x + 1$, what is f(g(2$$))$$?

Accepted Answer

$$9$$

Question 19

Which of the following is the correct set notation for the intersection of sets $$A$$ and $$B$$?

Accepted Answer

$$A \cap B = \{x \mid x 	ext{ belongs to both } A 	ext{ and } B\}$$

Study Notes: Numbers, Functions, and Foundations of Calculus (MATH 221 First Semester Calculus)

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