Domain: $$[-2, 2]$$, Range: $$[0, \\sqrt{8}]$$

$$(f \\circ g)(x) = \\sqrt{2x + 1}$$, domain: $$[-0.5, \\infty)$$

$$10 + \\ln(\\frac{e \\cdot 3}{6})$$

$$\\lim_{x o 2^-} f(x) = 3$$, $$\\lim_{x o 2^+} f(x) = 1$$

Question 1

Find the $$x-$$intercepts and $$y-$$intercept of the function $$y = x^2$ - 3x - 4.$$

Accepted Answer

$$x-$$intercepts: $$(-4, 0)$$ and $$(1, 0)$$; $$y-$$intercept: $$(0, -4)$$

Question 2

Find the domain and range of the function $$f(x) = \sqrt{8 - 2x^2$}.$$

Accepted Answer

Domain: $$[-2, 2]$$, Range: $$[0, \sqrt{8}]$$

Question 3

Given the function $$f(x) = 2x^2$ - 3x + 5$$, evaluate $$f(2).$$

Accepted Answer

$$7$$

Question 4

Simplify the difference quotient $$\frac{f(2+h) - f(2)}{h}$$ for $$f(x) = 2x^2$ - 3x + 5.$$

Accepted Answer

$$8 + 2h$$

Question 5

Use the Pascal Triangle to expand $$(A + B)^3$.

Accepted Answer

$$A^3$$ + $$3A^2B$$ + $$3AB^2$$ + $$B^3$$

Question 6

Given $$f(x) = \sqrt{x}$$ and $$g(x) = 2x + 1$$, write the expression for $$(f \circ g)(x)$$ and state its domain.

Accepted Answer

$$(f \circ g)(x) = \sqrt{2x + 1}$$, domain: $$[-0.5, \infty)$$

Question 7

Simplify $$10 + \ln 3 - \ln 6 + \ln e.$$

Accepted Answer

$$10 + \ln(\frac{e \cdot 3}{6})$$

Question 8

Evaluate $$\sin$$^{-1}$$(\frac{1}{2}).$$

Accepted Answer

$$\frac{\pi}{6}$$

Question 9

Using the technique of drawing a right triangle, find $$\sin(\cos$$^{-1}$$ x)$$ for $$0 \leq x \leq 1.$$

Accepted Answer

$$\sqrt{1 - x^2$}$$

Question 10

Given the graph of $$y = f(x)$$, determine $$\lim_{x 	o 2^-} f(x)$$ and $$\lim_{x 	o 2^+} f(x).$$

Accepted Answer

$$\lim_{x 	o 2^-} f(x) = 3$$, $$\lim_{x 	o 2^+} f(x) = 1$$

Question 11

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 12

Evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 13

Evaluate $$\lim_{x 	o 0} \frac{\tan x}{x}.$$

Accepted Answer

$$1$$

Question 14

Given $$\lim_{x 	o 0} f(x) = 5$$, $$\lim_{x 	o 0} g(x) = -2$$, and $$\lim_{x 	o 0} h(x) = 1$$, find $$\lim_{x 	o 0} (f(x)g(x) + h(x)).$$

Accepted Answer

$$-9$$

Question 15

Given the rational function $$f(x) = \frac{x^2$ - 2x}{x^2$ - 3x}$$, find the vertical asymptotes.

Accepted Answer

$$x = 0$$ and $$x = 3$$

Question 16

Evaluate $$\lim_{x 	o \infty} \frac{2x^2$ + 5}{x^2$ - 4}$$ and state if the function has a horizontal asymptote.

Accepted Answer

$$2$$, horizontal asymptote at $$y = 2$$

Question 17

Given $$f(x) = \frac{1}{x}$$, discuss the behavior of $$f(x) as$$ $$x$$ approaches $$0$$ from the right.

Accepted Answer

$$f(x) \to \infty$$

Question 18

If $$f(x) is $$not continuous at $$x = 3$$, which of the following could be a reason?

Accepted Answer

There is a jump discontinuity at $$x = 3$$

Foundations of Calculus: Functions, Limits, and Asymptotes

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