Question 1

Solve for $$x in $$the equation: $$\sqrt{512} = x^3$.

Accepted Answer

$$x = 8$$

Question 2

Given the function $$f(x) = 11$$^{x^3}$$ - 27$$, what is the domain and range of $$f(x)$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$; Range: $$(-27, \infty)$$

Question 3

For $$g(x) = \log(x + 3)$$, what is the vertical asymptote, domain, and range?

Accepted Answer

Vertical asymptote: $$x = -3$$; Domain: $$(-3, \infty)$$; Range: $$(-\infty, \infty)$$

Question 4

Which graph correctly represents $$f(x) = 2^x$$ and $$g(x) = \log_2$ x$$?

Accepted Answer

The exponential curve is increasing and passes through $$(0,1)$$; the logarithmic curve passes through $$(1,0)$$ and increases slowly for $$x \u003e 0.$$

Question 5

Evaluate $$\log_{10} 31 to $$four decimal places.

Accepted Answer

0.7646

Question 6

Solve for $$x$$: $$\log_4$ (x^2$ - 225) - \log_4$ (x + 15) = 1.$$

Accepted Answer

25

Question 7

Which graph represents $$f(x) = \left(\frac{1}{3}\right)^x$$?

Accepted Answer

A decreasing exponential curve passing through $$(0,1)$$ and $$(1,1/3).$$

Question 8

Does the line $$6x - 5y = 11$$ intersect the circle $$x^2$$ + $$y^2 = 49$?

Accepted Answer

Yes

Question 9

Write $$\frac{x^2$ - 2x + 1}{(x + 1)(x - 1)} in $$partial fraction decomposition form.

Accepted Answer

$$2/(x + 1) + 1/(x - 1)$$

Question 10

For the ellipse $$\frac{x^2$}{16} + \frac{y^2$}{49} = 1$$, what are the coordinates of its foci?

Accepted Answer

$$(0, -2\sqrt{33})$$ and $$(0, 2\sqrt{33})$$

Question 11

Given $$(y + 8)^2$ = 8(x + 6)$$, what is the vertex, focus, and directrix of the parabola?

Accepted Answer

Vertex: $$(-6, -8)$$; Focus: $$(-4, -8)$$; Directrix: $$x = -8$$

Question 12

Without completing the square, determine the type of conic section: $$121x^2$$ + $$49y^2 + 1452x - 1764y + 14303 = 0$.

Accepted Answer

Ellipse

Question 13

What type of graph does $$x^2$$ + $$y^2 = 147$ represent?

Accepted Answer

Circle

Question 14

Given the arithmetic sequence $$17, 9, 1, -7, \ldots$$, write the equation for the $$nth $$term and find $$a_{25}.$$

Accepted Answer

$$a_n = 17 - 8n; a_{25} = -183$$

Question 15

Evaluate $$\sum_{k=1}^{10} 2^k.$$

Accepted Answer

2046

Question 16

Apply the binomial expansion to $$(x + 13)^8$ and enumerate the first three terms.

Accepted Answer

$$x^8$$, $$8x^7 13$$, $$28x^6$$ $$13^2$$

Question 17

If the probability of a day being rainy is $$1/2$$, what is the probability that it will be sunny on eight successive days?

Accepted Answer

$$1/256$$

Question 18

For the sequence $$a_n = -13(n + 2)!$$, write its first four terms.

Accepted Answer

$$-78, -312, -1560, -9360$$

Comprehensive Study Notes for College Algebra Final Exam Topics

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