Stephen ARNOLD
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# Some Things to think About

### Investigating Functions and Graphs

#### Stephen Arnold

The following problems are chosen to encourage thinking, exploration and perhaps a few arguments!

1. How many functions can you find which are their own inverses?

How does a function affect its own inverse?

Discuss the domains and ranges of sin(arcsin(x)) and arcsin(sin(x)).

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3. How many functions can you find such that

f(x) + k = f(x + k)

for constant k?

(apart from f(x) = x)

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5. How do you define a perfect square?

Is 2x2 + 4x + 2 a perfect square?

Why or why not?

In how many different ways can a perfect square be defined?

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7. A popular method for solving equations involving absolute value is to square both sides.

Explain why this method works.

Will it always work?

In how many different ways can you solve

| | x | - 2 | = 1 ?

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9. The points A(-1, 1), B(1, 3), C(3, 1) and D(1, -1) form a quadrilateral.

In how many different ways can you prove that the quadrilateral is a rhombus?

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11. If x = 2 and x - 2 = 0 are equivalent mathematical statements, then why does squaring one give an equation with two distinct solutions, while squaring the other produces an equation with only one?

Now try to solve

x - sqrt( 6 - x ) = 0

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13. Study the graph of the composite function sin(tan(x)).

What is happening at each of the fuzzy bits?

If the function is periodic, then why do each of them appear different?

Now study the graph of the composite function F(x) = cos(arcsin(x))

Is it really a semi-circle, or does it just resemble one?

How might you prove this?

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15. Define a function R(x) as follows:

R(x) = R(x + 2) for all x

R(x) = x + 1 for 0 <= x < 2

What do you understand by the statement R(x) = R(x + 2)?

How would you explain this to someone else?

Can you describe the graph of y = R(x) and use it to find the value of R(4.5)?

(Barnes, 1988)

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17. If you are told that a function F(x) has F'(2) = 0 and F"(2) > 0, what can you deduce about F(2)?

What does it mean when f'(x) = f"(x) = 0?

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19. Suppose transportation specialists have determined that G(v), the number of litres of fuel per hundred kilometres that a vehicle consumes, is a function of the speed of the vehicle, in kilometres per hour.

Interpret, in terms of fuel consumption, the finding that G'(110) = 0.4.

How might this finding be used in a debate about setting an appropriate national speed limit? (Heid, 1988)

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21. E(t) represents the number of people unemployed in a country, t weeks after the election of a new government.

Translate each of the following facts about the graph, y = E(t), into statements about the unemployment situation:

1. The y-intercept of y = E(t) is 2 000 000.
2. E(20) = 3 000 000
3. The slope of y = E(t) at t = 20 is 10 000.
4. E"(36) = -800 and E'(36) = 0.
5. (Adapted from Heid, 1988)

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23. Solve: (x2 - 5x + 5)x2 - 9x + 20 = 1.

How many solutions did you find?

Are you sure you have them all?

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For comments & suggestions, please e-mail Steve Arnold.