The notation f (x) = 5x2 - 1 is a convenient way to represent a function.
How do we read and say the symbols f (x)?
We say "f of x".
Do the parentheses in "f (x)" mean multiplication?
No.
Are the parentheses grouping symbols here?
No.
So, "f (x)" (is / is not) the product of "f" and "x".
"f (x)" is not the product of "f" and "x".
What does "f (x)" mean?
"f (x)" denotes the value of the function "f" at "x".
When we write f (x) = 5x2 - 1, what is the independent variable?
x is the independent variable.
What is the name of this function?
The name of the function is "f".
To summarize, in "f (x)", _____ is the name of the function, and x is the ____________.
To summarize, in "f (x)", "f" is the name of the function, and "x" is the independent variable.
What is the definition of this function?
"f" is defined as "5x2 - 1"
In f (x) = 5x2 - 1, the expression on the right side can be thought of as a set of instructions describing what to do with the variable x.
Express those instructions in words.
f (x) = 5x2 - 1 means: Given a value of x, you must square it, multiply by 5, and subtract 1.
Suppose that the value of x is given as 3.
That is, f (3) = 5(3)2 - 1.
State the instructions for this case.
f (3) means that you must square the 3, multiply by 5, and subtract 1.
Notice the parentheses on the right side of
f (3) = 5(3)2 - 1.
It helps to use similar parentheses in f (x).
Try it.
f (x) = 5(x)2 - 1.
An advantage of this is that it emphasizes that whatever is in the parentheses on the left, we square it, multiply by 5, and subtract 1.
Now express f (x+4) in symbols.
f (x+4) = 5(x + 4)2 - 1
Do the multiplication.
We expand the square of (x+4) by expressing it as a product:
f(x+4) = 5(x+4)(x+4) - 1. Applying FOIL, we get
f(x+4) = 5(x2 + 8x + 16) - 1, or
f(x+4) = 5x2 + 40x + 80 - 1, or
f(x+4) = 5x2 + 40x + 79.
The value of the independent variable may be a fraction. Express 
in symbols.
Do we need to square the "3" as well as the "a"?
Yes.
Do the multiplication.
A value of the independent variable which appears frequently in later math courses is (x+h).
Express f (x + h) in symbols.
f (x + h) = 5(x+h)2 - 1
Do the math.
Expand the square as a product:
f (x + h) = 5(x+h)2 - 1 = 5(x+h)(x+h) - 1
= 5(x2 + 2xh + h2) - 1
= 5x2 + 10xh + 5h2 - 1.
Now we'll use function notation to talk about points on the graph a graph. Use Figure 1as an example. What is the meaning of "f (2)" with "f" referring to the graph?
It means to read the value of "f(x)" at the location where x = 2.
Find f(2).
Since the y coordinate has the value -2 at x = 2,
we can say that
the point being referenced is (2, -2).
Consequently, f(2) = -2.
Let's try another value of x.
Evaluate f (-5).
f (-5) = 1.
Try one more: Find f (6).
f (6) = 0
What does f (x) = 2 mean?
It describes a point with a y coordinate of 2 and an x-coordinate which we can find from the graph.
Solve f (x) = 2.
From the graph, we can see two places where the y-coordinate ( = f (x) ) is equal to 2. This occurs at x = -6 and x = 0.
Solve f (x) = 1.
There are four places where f(x) = 1. These occur at x values of -5, -1, 1, and 8.
