**Q: Find the following values of the function: f(x) = 3x + 1**

** a. f(0) b. f(3) **

**c. f(-2) d.f(-x) **

**e. -f(x) f. f(x+2) **

**g. f(2x) h. f(x+h)**

A: As I’ve mentioned before, a function is just a machine. You put something in it (a variable, a number, etc…) and something else comes out (maybe a different number or a different variable). So, let’s start:

**(a) **f(x) = 3x + 1

The red parts will always match. So, f(7) = 3(7) + 1… Therefore, f(0) means to replace all of the x’s in the equation with 0’s:

f(0) = 3(0) + 1 = 0 + 1 = 1

**(b)**

f(3) = 3(3) + 1 = 9 + 1 = 10

Again, see how the “3” just replaced the x?

**(c)**

f(-2) = 3(-2) + 1 = -6 + 1 = -5

**(d) **Find f(-x). This is a little more confusing to look at, but no different than before. The “-x” just replaces all of the x’s:

f(-x) = 3(-x) + 1 = -3x + 1

**(e)** -f(x) = ?? Notice there is just a negative sign in front of f(x)? So start with:

f(x) = 3x + 1

Now, add a negative in front of both sides:

-f(x) = -(3x + 1)

-f(x) = -3x – 1

TADA!

**(f) **Back to just plugging in to the function:

f(x + 2) = 3(x + 2) + 3 = 3x + 6 + 3 = 3x + 9

**(g) **Plug in again:

f(2x) = 3(2x) + 1 = 6x + 1

**(h) **More with the plugging in:

f(x + h) = 3(x + h) + 1

f(x + h) = 3x + 3h + 1

Wow. That was all of them.

Remember…. A function is a set of rules… f(x) is read as “f of x”: meaning that the function f depends on the input x.

f(3) is read as “f of 3” and it means to plug “3” into the function.

Please notice the difference between the two:

f(x + h) and f(x) + h

Notice that the problem in read has (x + h) plugged into the function…. Therefore, using our function f(x) = 3x + 1, that would make:

f(x + h) = 3(x + h) + 1

Notice that the (x + h) went *into* the function replacing the original x.

Now, consider the problem in blue : f(x) + h. This is the function f(x) with an h added to the end. So, if f(x) = 3x + 1, then:

f(x) + h = (3x + 1) + h

See the difference? This is a very important concept to understand about functions.