Week 4: Functions – Answers

Week 4: Functions – Day 5

1. What is the full domain of f? (Do you know approximately how big this domain is?)

If the input is human beings who live on Earth, the full domain is the set of all human beings who live on Earth. This domain is finite, and has about 7 billion elements (in 2014).

2. Is f(x) a bounded function? If so, what are the bounds of f(x)?

f(x) is bounded. f(x) is bounded below by zero, and there exists an absolute minimum of f(x) (after checking the heights of all 7 billion people, a minimum could be calculated). The function is bounded above by human physiology. There are no recorded instances of a human being over 9 feet (244 cm) tall.

3. From the above data, give an example of a local maximum of f(x). Specify what the value of the local maximum is, and where it occurs, as best you can.

Examples: local maximum at Palo Alto, where the value is 175 cm.

Other examples: North America, value of 190 cm; Africa, 199 cm; Australia, 198 cm; Antarctica, 165 cm.

4. In your specific example of a local maximum of f(x) in problem 3, what is the subdomain where the local maximum occurs?

The first subdomain mentioned was Palo Alto.

5. In the above data, if the domain is restricted to thirteen year old girls, does f(x) have an absolute maximum?

It looks like the absolute maximum on the subdomain of thirteen year old girls is 200 cm, which is achieved somewhere in Europe or Asia.


 

Week 4: Functions – Day 4

For each of the functions below, calculate the average growth rates on two intervals: [0, 1] and [1, 2]. First think: what do you expect the answers to be. Do the answers fit your expectations?

1. f(x) = 2x + 5

On the interval [0, 1], the average growth rate is: (2×1 + 5 – (2×0 + 5))/ (1 – 0) = (7 – 5)/1 = 2.

On the interval [1, 2], the average growth rate is: (2×2 + 5 – (2×1 + 5))/(2 – 1) = (9-7)/1 = 2.

The two growth rates are the same, 2. This is what we expect because this function describes a straight line relationship between x and y. Straight lines have constant (unchanging) growth rates.

2. f(x) = |x – 1|

On [0, 1]: average growth rate = (|1 – 1| – |0 – 1|) / (1-0) = (0 – 1)/1 = -1.

On [1, 2]: average growth rate = (|2 – 1| – |1 – 1|) / (2-1) = (1 – 0)/1 = 1.

The two average growth rates are -1 and 1. This makes sense because this function is an absolute value with minimum point at x=1. To the left of x=1, the function decrease with slope -1. To the right of x=1, the function increases with slope 1.

3. f(x) = x – 1

On [0, 1]: average growth rate = ( (1-1) – (0-1)) /(1-0) = 1/1 = 1.
On [1, 2]: average growth rate = ( (2-1) – (1-0)) /(2-1) = 1/1 = 1.

Both rates are the same, 1. This is expected because this function is has a straight line graph.

4. f(x) = sqrt(x) (the square root of x). For this problem, take the square root of 2 to be 1.414.

On [0, 1]: average growth rate = (sqrt(1) – sqrt(0)) /(1-0) = 1/1 = 1.

On [1, 2]: average growth rate = (sqrt(2) – sqrt(1)) /(2-1) = (1.414 – 1.0)/1 = 0.414

5. f(x) = -(x – 1)2

n [0, 1]: average growth rate = (-(1 – 1)2 – -(0 – 1)2) /(1-0) = 1/1 = 1.

On [1, 2]: average growth rate = (-(2 – 1)2 – -(1 – 1)2) /(2-1) = -1/1 = -1.

This is expected because the graph of f(x) is an inverted parabola with maximum point at x=1. To the left of x=1, the function increases. To the right of x=1, the function decreases.


 

Week 4: Functions – Day 3

For each function below state whether it is increasing, decreasing, or staying the same on the specified domain. (Hint: the functions below do not wiggle around a lot. In the specified domain they are either increasing, decreasing or staying the same).

1. f(x) = 5x – 6 domain: all x

The graph of this function is a straight line with positive slope. It rises steadily (increases).

2. f(x) = 19 domain: all x

The graph of this function is a horizontal line. The value of the function stays the same.

3. f(x) = -3x + 75 domain: all x

The graph of this function is a straight line with negative slope. It decreases.

4. f(x) = x2 domain: x is > 0.

For x > 0, f(x) increases.

5. f(x) = 1/x domain: x is not equal to 0.

For all x not equal to zero, f(x) decreases. (When x is negative, f(x) becomes more negative as x increases. When x is positive, f(x) is positive but approaches zero.) Convince yourself using a value chart:

x = -3, y = -1/3
x = -2, y = -1/2 (more negative)
x = – 1, y = -1 (more negative)
x = 1, y = 1
x = 2, y = 1/2 (decrease from x = 1 to x = 2)
x = 3, y = 1/3 (decrease from x = 2 to x = 3)
If you plug in many values of x you can see that these trends are followed for numbers in between. You can also graph this function to see the trend.


 

Week 4: Functions – Day 2

If a function’s range has some limits, the function is bounded. If the function’s range has no limits, the function is unbounded. Are these functions bounded or unbounded? If bounded, do they have an upper bound, lower bound, or both?

1. f(x) = 3x -9

This function ranges over all numbers. It has no bounds. It is unbounded.

2. f(x) = -x2

This function has a maximum value of zero, since all other values are negative (the x2 is always positive). This function can have arbitrarily large negative values.

Therefore, this function has an upper bound of zero, and no lower bound.

3. f(x) = |x – 2|

This function has a minimum value of zero, and no maximum. It has a lower bound and no upper bound.

4. f(x) = x3

This function ranges over all numbers. It is unbounded.

5. f(x) = 1/(x2 + 1)

This function is bounded. It has both upper and lower bounds.

This function has a numerator of 1 and a denominator that is greater than or equal to 1. That means that the value of the function is always less than or equal to 1. The value 1 is an upper bound and maximum.

The numerator of f(x) is never zero, and so f(x) is never zero. The denominator of f(x) is always positive, (and the numerator), so f(x) is never negative. When the denominator is a very large number (when x is very large) the value of f(x) is close to zero. Zero is a lower bound for f(x). f(x) does not have a minimum value.


 

Week 4: Functions – Day 1

What are the domain and range of each function?

1. f(x) = |x|

Domain: all numbers
Range: y >= 0

2. f(x) = 2x – 3

Domain: all numbers
Range: all numbers

3. f(x) = x – 4

Domain: all numbers except zero
Range: all positive numbers (y > 0)

4. f(x) = 1 – |x|

Domain: all numbers
Range: y <= 1 (all values less than or equal to 1)

5. f(x) = (1 – |x|)2

Domain: all numbers
Range: because the function has a square, it cannot be negative. It can reach zero. The range is y >= 0 (all nonnegative numbers).

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