1) A function relates an input value x to an output value y or f(x). For example, in the function f(x) = x - 5, f(12) would evaluate to 7. 2) Functions are often written using function notation like f(x) instead of y. This notation means "function of x". 3) Evaluating a function means replacing the x values in the function with numbers and calculating the output. For example, evaluating f(x) = x^2 - 3x at x = -8 would give an output of 88.

1. Skill40 Evaluating Functions
One way to think about a function is that it is like a machine when you put something
in, say a number x, and then the machine “works” on the number. Then, it spits
out a number y. The “x” number is the input while the “y” number is the output.
We sometimes use a different kind of notation to say this. However, the idea is still
like saying y = x -5 so find y when x = 12. You would just think y = 12 – 5 or y = 7.

2. In function notation we write f(x) instead of y but we still mean what
number you get when you replace the x in an equation with the number
inside the f(x) parentheses.
The notation just means literally “function of x.” We often use f(x) just
because the word function starts with an f. However, we can use any letter.
Examples: g(x) or h(x) or even h(t). Sometime we don’t even use an x. In
science, we like to use letters that start the words the letters stand for like
h(t) for a height at time t.
So, y = x – 5 would look like f(x) = x – 5 . We would say find f(12) when we
mean to replace the x on the right side with 12. We call this “evaluating a
function.”

3. Here are two problems that show how to think about evaluating a function.
The first one is a “linear” function. That means its graph would be a straight line.
The second one is a “quadratic function.” Its graph would be a curve. Notice the
X squared.
Just put 2 in for x to see what
you get.

4. In the book, pages 191 and 192 have puzzle problems that give practice in
evaluating a function. However, the puzzles are a little confusing so here are
some practice problems.
g(x) = 3x − 3; Find g(−6) So, g(-6) = 3(-6) -3
g(-6) = -18 – 3
and g(-6) = -21
f (x) = x2 − 3x; Find f (−8) So, f(-8) = (-8)2 -3(-8)
f(-8) = 64 +24 You can put the right side in a calculator.
f(-8) = 88
h(n) = −2n2 + 4; Find h(4) h(4) = -2(4)2 + 4
h(4) = -2 (16) + 4
h(4) = -32 + 4
h(4) = -28 We use the order of operations PEMDAS.

5. Be ready to evaluate a function for a fraction or decimal. The idea is
the same and you can use a calculator. Change fractions to decimals
if you can.
The x and y or x and f(x) in function problems are just like (x,y) ordered pairs that
you have studied before.
If I tell you that f(2) = 8, that is like telling you that x = 2 and y = 8 or (2,8) is a point
on a graph.

6. In mathematics, we like to think about functions in more than one way (multiple ways).
Here is a function described by an equation, f(x) = 2x + 1. We can make a table of
values for the function by replacing x with each of the numbers in the chart one at a
time.
X -3 -2 0 1 2
F(x) 2(-3) +
1 or -5
2(-2) +
1 or -3
2(0) + 1
or 1
2(1) + 1
or
3
2(2) + 1
or 5
We often write the tables vertically so that it might
look like this. Then we can plot these as (x,y) points on
the graph. Then join them with a line. This is messy I
know! The line should go through the points. Sorry.
X
-3
-2
0
1
2
F(x)
-5
-3
1
3
5