# Introduction to Functions

©2018 Art Middlekauff

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:

Ask: What observations can you make about these two lines?

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.