Q: A real-world application: The height $$h (in $$meters) of a ball thrown upward from the ground at time $$t (in $$seconds) is given by $$h(t) = -4.9t^2$ + 20t. $$What is the height of the ball after 2 seconds?

$$h(2) = -4.9 \times 4 + 20 \times 2 = -19.6 + 40 = 20.4$$ meters

Q: Suppose $$f(x) = 2x + 1$$ and $$g(x) = x^2$. What is (f$$ \circ g)(2)$$?

$$(f \circ g)(2) = f(g(2)) = f(4) = 2 \times 4 + 1 = 9$$

Question 1

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

Accepted Answer

$$\{x \mid 1 \u003c x \u003c 6\}$$

Question 2

Which of the following numbers is NOT a rational number?

Accepted Answer

$$\sqrt{2}$$

Question 3

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

Accepted Answer

$$\{x \mid x 
eq 2\}$$

Question 4

Which of the following statements about the graph of a function is TRUE?

Accepted Answer

The graph passes the vertical line test if and only if it represents a function.

Question 5

Suppose you are measuring the distance between two points $$a$$ and $$b on $$the real number line. What is the formula for the distance?

Accepted Answer

$$|a-b|$$

Question 6

Which of the following is the correct definition of a function $$f$$ from a set $$A to a $$set $$B$$?

Accepted Answer

A rule which assigns to each element of $$A$$ exactly one element of $$B.$$

Question 7

If $$f(x) = mx + b is a $$linear function, what does the constant $$m$$ represent?

Accepted Answer

The slope of the line

Question 8

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

Accepted Answer

$$[-2, 5)$$

Question 9

A student claims that the decimal $$0.333... is a $$rational number. Is this correct, and why?

Accepted Answer

Yes, because $$0.333... = \frac{1}{3}$$, which is rational.

Question 10

Which of the following sets is the solution to $$x^2 = 4$?

Accepted Answer

$$\{-2, 2\}$$

Question 11

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

Accepted Answer

$$\{x \mid x \geq 0\}$$

Question 12

Which of the following is an example of a function that is NOT defined for all real numbers?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 13

A real-world application: The height $$h (in $$meters) of a ball thrown upward from the ground at time $$t (in $$seconds) is given by $$h(t) = -4.9t^2$ + 20t. $$What is the height of the ball after 2 seconds?

Accepted Answer

$$h(2) = -4.9 	imes 4 + 20 	imes 2 = -19.6 + 40 = 20.4$$ meters

Question 14

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

Accepted Answer

$$\mathbb{Q}$$

Question 15

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

Accepted Answer

$$\{y \mid y \geq 0\}$$

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

Suppose $$f(x) = 2x + 1$$ and $$g(x) = x^2$. What is (f$$ \circ g)(2)$$?

Accepted Answer

$$(f \circ g)(2) = f(g(2)) = f(4) = 2 	imes 4 + 1 = 9$$

Question 18

Which of the following is the correct way to represent the set of all real numbers except $$0$$?

Accepted Answer

$$\mathbb{R} \setminus \{0\}$$

Question 19

A chemical reaction rate $$R$$ depends on temperature $$T$$ according to $$R(T) = kT^2$, where k$ is a constant. If the temperature doubles, how does the reaction rate change?

Accepted Answer

The reaction rate increases by a factor of 4.

Numbers, Functions, and Foundations of Calculus

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