Functions

Preparation: From our syllabus...

You are expected to come prepared. This means that you should have read the sections to be covered and should be prepared to ask questions about the reading and the problems you are asked to do.

Though, for this first class of the semester I do not expect you to have started this habit yet!

Research has shown that "active learning" in which students spend class time engaging with material, is more effective than simply hearing about content. Instead of imparting information in lectures, class activities will be focussed on applications and implications. And this means you need to get enough of the content ahead of time to engage.

Functions [1.1]

A function is:

• a rule that takes certain numbers as inputs and assigns to each a definite output.
• The set of all input numbers is called the domain of the function.
• The set of all resulting output numbers is called the range.
  
  

Example

• Rule: What was the maximum temperature, $T$ (measured in degrees F) at the Goshen airport on a particular date?
• Input / "independent" variable: date, "$d$", e.g. August 4, 2014.
• output / "dependent" variable: maximum temperature, $T$, e.g. 82 F.
  
  

Function notation: $T(d)$
What might be the domain of this function?

Identical functions or not?

Consider the two functions, defined by $f(x)=\sin(x)+2\log(x)$ and $g(u)=\sin(u)+2\log(u)$. Which is true?

1. $f$ and $g$ are exactly the same functions.
2. If $u$ and $x$ are different numbers, then $f$ and $g$ are different functions.
3. Not enough information is given to determine if $f$ and $g$ are the same.

Let $f(x)=(x^2-4)/(x-2)$ and $g(x)=x+2$.

Are $f$ and $g$ the same functions?
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No: For almost all input numbers the two functions give the same output number. But there is one exception: $g(2)=4$, but $f(2)= 0/0$ which is not the same as 4. So, they do not always give the same output for a given input.

Interpreting function notation

A patient’s heart rate, $R$, in beats per minute, is a function of the dose, $D$ of a drug, in mg. We have $R = f(D)$. The statement $f(50) = 70$ means:

1. The patient’s heart rate goes from 70 beats per minute to 50 beats per minute when a dose is given.
2. When a dose of 50 mg is given, the patient’s heart rate is 70 beats per minute.
3. The dose ranges from 50 mg to 70 mg for this patient.
4. When a dose of 70 mg is given, the patient’s heart rate is 50 beats per minute.

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b.

Representations of functions

• Formula.
• Statement in words.
• Graph of a relationship.
• Table of values (NOAA data).

A table

As a person hikes down from the top of a mountain, the variable $T$ represents the time, in minutes, since the person left the top of the mountain, and the variable $H$ represents the height, in feet, of the person above the the bottom end of the trail. Here are some values at several different times for these variables:

Which of these statements is true?

1. $T$ is a function of $H$.
2. $H$ is a function of $T$.
3. Both of a) and b) are true.
4. Neither a) nor b) are true.

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b. is correct. "a." is not possible, because if $H$ were the input, then $T$ would be the output. But there are two entries in the table that have $H=810$: One has $T=30$ and one has $T=50$. So, there would be no unique output for the input value 810.

Vertical intercept

The vertical intercept of a function, $f(x)$ is the value of the function when $x=0$.

Or... $f(0)$.

Or... The value of the function, where its graph crosses the $y$-axis (a vertical lineat $y=0$).

As a person hikes down from the top of a mountain, The variable $T$ represents the time, in minutes, since the person left the top of the mountain. And the variable $H$ represents the height, in feet, of the person above the base of the mountain. We have $H=f(T)$. The vertical intercept for the graph of this function represents:

1. The time it takes the person to descend from the top of the mountain to the base of the mountain.
2. The height of the person in feet above the base of the mountain when the person is at the top of the mountain.
3. The height of the person in feet above the base of the mountain, as the person hikes down the mountain.
4. The time when the person begins to descend down the mountain.

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The vertical intercept is the value of the function when the argument, $T$, is zero. $f(0)$ represents the height of the person at time $T=0$. According to the problem, at $T=0$ the person is at the top of the mountain. So the answer is b.

Horizontal intercept

The horizontal intercept of a function, $f(x)$ is the value of $x$ such that $f(x)=0$.

Or... The value of $x$ when the graph of a function crosses the $x$-axis (a horizontal line).

As a person hikes down from the top of a mountain, The variable $T$ represents the time, in minutes, since the person left the top of the mountain. And the variable $H$ represents the height, in feet, of the person above the base of the mountain. We have $H=f(T)$. The horizontal intercept for the graph of this function represents:

1. The time it takes the person to descend from the top of the mountain to the base of the mountain.
2. The height of the person in feet above the base of the mountain when the person is at the top of the mountain.
3. The height of the person in feet above the base of the mountain, as the person hikes down the mountain.
4. The time when the person begins to descend down the mountain.

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The horizontal intercept is the value of $T$ when the function's value is 0. $H=0$ corresponds to the base of the mountain. The answer is a.

Increasing and decreasing function

A function $f$ is increasing if the output values, $f(x)$ are increasing as the input values, $x$, increase.

A function $f$ is decreasing if the output values, $f(x)$ are decreasing as the input values, $x$, decrease.

A patient's heart rate, $R$, in beats per minute, is a function of the dose, $D$ of a drug, in mg. We have $R=f(D)$. The vertical intercept for the graph of this function represents:

1. The maximum dose of the drug.
2. The maximum heart rate.
3. The dose of the drug at which the patient's heart stops beating.
4. The patient's heart rate if none of the drug is administered.

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The vertical intercept occurs when the argument, $D$, is 0. That is, when the dose is zero. The answer is d..

Sketch a plausible graph of this function, compatible with the two conditions above: If the patient's heart rate without taking any of the drug is 60 bpm, and the function is increasing.

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Your graph should have a vertical intercept of 60, and and increase as $D$ gets larger (as you go to the right).

Graphs

In which of these graphs could $y$ be a function of $x$?

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In both II and IV there are $x$ values for which there are more than one possible $y$-value. (That is, a vertical line would cut the graph at more than one place.) So the answer is I or III.

Have you heard of the vertical line test?

Motion

Sketch a plausible graph of Joey's distance, $d$, from pole A as a function of time, $t$.

Which of these graphs best captures this motion?

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Joey starts (time=0) at pole C which is a distance of 10 meters from the origin. So, that rules out graphs a and c, because they have a vertical intercept of 0.

A steep line means a big change in distance over a period of time (fast motion). A shallow line means a smaller change in distance over a period of time (slow motion). The answer is d. The graph starts shallow (slow), horizontal (stopped), then steep (fast).

Domains and ranges

Which of the following functions has the same domain as its range?

1. $f(x)=x^2$
2. $g(x)=\sqrt x$
3. $h(x)=x^3$
4. $i(x)=|x|$

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The answer is b (range and domain are (0,$+\infty$)) and c (range and domain (-$\infty$ , +$\infty$)).