Section 1.1 Variables and functions

Before we answer that question, are you wondering why there’s that extra 9 at the end of the price? We might think a gallon costs $3.99 but there’s really a small 9 following it. Sometimes that 9 is raised up slightly on the gas station sign. You have to read the fine print. What it means is an extra \(\frac{9}{10}\)¢ for each gallon. So the true price of a gallon of gas would be $3.999. Gas costs a tiny bit more than you thought. Good grief.

Back to our question. Does 70¢ truly matter to us? Probably not. Can’t even buy a bag of potato chips for 70¢. But, how often do you buy just one gallon of gas? Typically you might put five, or ten, or even twenty gallons of gas into the tank. We want to understand how the price of gasoline influences what it really costs us at the pump.

To do that let’s compare our costs when we buy ten gallons of gas. This is a new quantity in our story, but it’s one that we won’t change - it’s going to be constant throughout our story. There’s no particular reason for picking ten; it’s just a nice number to work with.

In talking about the relationship between these variables we might say “the total cost depends on the price of gas,” so \(C\) depends on \(P\text{.}\) That tells us that \(C\) is the dependent variable and \(P\) is the independent variable. In general, the variable we really care about is the dependent variable, in this case \(C\text{,}\) the total amount of money it costs us. The concept of dependence is so important that there’s yet another word for it. We say that \(C\) is a function of \(P\text{,}\) as in “cost is a function of price.”

Given a choice, we usually assign dependence such that given a value of the independent variable, it is easy to calculate the corresponding value for the dependent variable. In our example it’s easy to use the price per gallon, \(P\text{,}\) to figure out the total cost, \(C\text{.}\) We can work backwards - from \(C\) to \(P\) - but it’s not as easy.

From our experience we have a sense of what gas might cost. In my lifetime, I’ve seen gas prices as low as 35.9¢/gallon in the 1960s to a high of $4.099/gallon recently. This range of values sounds too specific, so it would sound better to say something general like

“Gas prices are (definitely) between $0/gal and $5/gal.”

Be aware that there are often many different numbers in a story. Some numbers are examples of values the variables take on, such as $3.999/gal or $39.99 in our example. Other numbers are values of quantities that are constants; they do not change (at least not during the story). The one constant in our story is that we are always buying 10 gallons of gas. Occasionally there are other numbers in a story that turn out not to be relevant at all, so be on the lookout.

Back to our story. A report says that the average price of gasoline in Minnesota was $2.900/gal in 2010 and increased approximately 20% per year for the next several years. We would like to check what that says about the average price of gasoline in 2011 and 2012, say. (It is unlikely that the price increase continued much longer at that rate.)

One last note. The number 20% in the report sounds like a rough approximation. The report probably means the increase was around 20%, maybe a little less, maybe a little more. So our answers of $3.48/gal and $4.176/gal could be a little less or a little more too. But they sound so perfectly correct. To be safe, we really ought to round off these answers, to something more general like around $3.50/gal in 2011 or approximately $4.20/gal in 2012. Using our “approximately equal to” symbol we write \(P \approx \$3.50\)/gal in 2011 and \(P \approx \$4.20\)/gal in 2012.

If you’re not sure, work the rest of the exercises and then return to these questions. Or, ask your instructor or a classmate for help.