Calculus/Precalculus/Solutions

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Convert to interval notation[edit | edit source]

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This is equivalent to
This is equivalent to
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It helps to draw a picture to determine the set of numbers described:
5 lt x or x lt 6.png

A number in the set can be on either the red or blue line, so the entire number line is included.

It helps to draw a picture to determine the set of numbers described:
5 lt x or x lt 6.png

A number in the set can be on either the red or blue line, so the entire number line is included.

State the following intervals using set notation[edit | edit source]

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Which one of the following is a true statement?[edit | edit source]

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Let . Then

, and

Thus,

false
Let . Then

, and

Thus,

false
18.
Using the same example as above, we have .
false
Using the same example as above, we have .
false
19.
true
true

Evaluate the following expressions[edit | edit source]

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Simplify the following[edit | edit source]

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Find the roots of the following polynomials[edit | edit source]

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Factor the following expressions[edit | edit source]

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Simplify the following[edit | edit source]

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Functions[edit | edit source]

52. Let .

a. Compute and .
,
,
b. What are the domain and range of ?
The domain is ; the range is ,
The domain is ; the range is ,
c. Does have an inverse? If so, find a formula for it.
No, since isn't one-to-one; for example, .
No, since isn't one-to-one; for example, .

53. Let , .

a. Give formulae for
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provided . Note that 0 is not in the domain of , since it's not in the domain of , and you can't divide by something that doesn't exist!
provided . Note that 0 is not in the domain of , since it's not in the domain of , and you can't divide by something that doesn't exist!
vi.
. Although 0 is still not in the domain, we don't need to state it now, since 0 isn't in the domain of the expression either.
. Although 0 is still not in the domain, we don't need to state it now, since 0 isn't in the domain of the expression either.
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b. Compute and .
; .
; .
c. Do and have inverses? If so, find formulae for them.
Yes; and . Note that and its inverse are the same.
Yes; and . Note that and its inverse are the same.
54. Does this graph represent a function? Sinx over x.svg
As pictured, by the Vertical Line test, this graph represents a function.
As pictured, by the Vertical Line test, this graph represents a function.

55. Consider the following function

a. What is the domain?
b. What is the range?
c. Where is continuous?

56. Consider the following function

a. What is the domain?
b. What is the range?
c. Where is continuous?

57. Consider the following function

a. What is the domain?
b. What is the range?
c. Where is continuous?

58. Consider the following function

a. What is the domain?
b. What is the range?
c. Where is continuous?

Graphing[edit | edit source]

59. Find the equation of the line that passes through the point (1,-1) and has slope 3.
60. Find the equation of the line that passes through the origin and the point (2,3).