Introduction to Functions

Introduction to Functions

For personal home schoolroom use only. Permission must be granted by the author for any other usage. Adapted from Paul A. Foerster’s Algebra and Trigonometry, section 4-4, pp. 126-133 (Pearson Prentice Hall, 2006).

Show the student two equations:

$y = \frac{3}{4}x+3$

$y = \frac{5}{2}x-5$

Ask the student to graph these two lines on a single Cartesian coordinate system. Here is an example of what the student would draw:

What do they have in common? (possible answer: they have a positive slope)
How are they different? (possible answers: slope, intercepts)
Which is above the other?
Which is steeper?

What does y equal when x is 4? (help the student realize that this is an ambiguous question, since there are two lines)

How can we specify which y we want?
Do you have ideas how we could specify which line we are referring to?

What if we labeled the first one line f and the second one g?

Which equation has the steeper slope? (f or g?)
Which equation has a positive y-intercept? (f or g?)
What does y equal when x is 4, for f?
What does y equal when x is 4, for g?

Explain: Mathematicians use terminology and notation to differentiate between equations, or functions, with the same dependent variable. They write:

$f(x) = \frac{3}{4}x+3$

$g(x) = \frac{5}{2}x-5$

Instead of saying y equals 6 when x is 4, for f, they write:

$f(4) = \frac{3}{4}\cdot4+3 = 3 + 3 = 6$

And instead of saying y equals 5 when x is 4, for g, they write:

$g(4) = \frac{5}{2}\cdot4-5 = 10 - 5 = 5$

This notation was first used by Leonhard Euler in 1734, one of the most prolific mathemeticians in history.

f and g are called functions.

Can you think of any benefits of this function notation?

Let’s see what we can do with it. Here are two new function definitions:

$f(x) = 3x + 11$

$g(x) = x^2 + x + 1$

Evaluate together:

$f(7)$

$g(-3)$

$\frac{f(5)}{g(5)}$

$f(g(2))$

Evaluate alone:

$f(0)$

$g(\frac{1}{3})$

$\frac{g(1)}{g(0)}$

$g(g(0))$

Explain: One reason function notation is so powerful is that often functions have relationships between each other.

A large part of calculus involves relationships between functions.

This is especially true for calculus that applies to real world processes such as motion and acceleration.

Function notation give us a powerful and expressive way to explore these relationships, so that we can better understand dynamic processes in the real world.