Q: Given the function $$f(x) = \frac{1}{x-2}$$, what is the domain of $$f$$?

$$\{x \in \mathbb{R} \mid x \neq 2\}$$

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

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

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

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

Accepted Answer

The set of all real numbers $$x$$ for which $$f(x) is $$defined.

Question 3

Given the function $$f(x) = \frac{1}{x-2}$$, what is the domain of $$f$$?

Accepted Answer

$$\{x \in \mathbb{R} \mid x 
eq 2\}$$

Question 4

A function $$f is $$given 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 5

Which of the following is NOT a property of real numbers?

Accepted Answer

Every real number can be written as a finite decimal.

Question 6

Suppose $$f(x) = \sqrt{x-1}. $$What is the domain of $$f$$?

Accepted Answer

$$\{x \in \mathbb{R} \mid x \geq 1\}$$

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 the distance between two numbers $$a$$ and $$b on $$the number line is $$|a-b|$$, what is the distance between $$-3$$ and $$5$$?

Accepted Answer

$$8$$

Question 9

Which of the following is the correct set notation for the interval containing all real numbers $$x$$ such that $$2 \leq x \u003c 5$$?

Accepted Answer

$$[2, 5)$$

Question 10

A real-world application: The distance $$d$$ between two cities on a straight road is $$120 km. If a $$car travels from city $$A to $$city $$B at a $$constant speed of $$60 km/h$$, how long does the trip take?

Accepted Answer

$$2$$ hours

Question 11

Which of the following is the correct definition of a function?

Accepted Answer

A rule which tells you how to compute the value $$f(x) of $$the function for a given real number $$x.$$

Question 12

Which of the following is the range of the function $$f(x) = \sqrt{x}$$ for $$x \geq 0$$?

Accepted Answer

$$[0, \infty)$$

Question 13

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

Accepted Answer

$$11$$

Question 14

Which of the following numbers is NOT a rational number?

Accepted Answer

$$\sqrt{2}$$

Question 15

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

Accepted Answer

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

Question 16

Which of the following is the correct interval notation for all real numbers $$x$$ such that $$x \u003e 3$$?

Accepted Answer

$$(3, \infty)$$

Question 17

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

Accepted Answer

$$9$$

Question 18

Which of the following statements about the number $$\pi is $$true?

Accepted Answer

$$\pi is an $$irrational number.

Question 19

A function $$f is $$defined by $$f(x) = \frac{1}{x}. $$Which of the following statements is true?

Accepted Answer

The domain of $$f is $$all real numbers except $$0.$$

Numbers, Functions, and Foundations of Calculus

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