If f (x) = 3x + 4

Function problems, like f (x), g (x), h (x), etc, leave a lot of students confused. This is unfortunate, because function problems are nothing more than substitution problems. I'll give you an example:

f (x) = 3x + 4
f (2) = ?

We note that in left half of the question, 2 has been plugged in where x was. To solve the problem, then, all we need to do is substitute (plug in) 2 for x in the right half of the equation:

f (2) = 3(2) + 4 = 6 + 4 = 10

Similarly,

f (3) = 3(3) + 4 = 9 + 4 = 13
f (4) = 3(4) + 4 = 12 + 4 = 16
f (5) = 3(5) + 4 = 15 + 4 = 19
f (6) = 3(6) + 4 = 18 + 4 = 22
f (7) = 3(7) + 4 = 21 + 4 = 25
f (8) = 3(8) + 4 = 24 + 4 = 28
f (9) = 3(9) + 4 = 27 + 4 = 31
f (10) = 3(10) + 4 = 30 + 4 = 34
f (y) = 3(y) + 4 = 3y + 4
f (z) = 3(z) + 4 = 3z + 4
etc.

Sometimes, we might find multiple functions used together. When this is the case, we just follow the usual rules of math to untangle the question:

if f (x) = 3x + 4, and g(x) = 5x - 7
f (4) - g (2) = ?

As before, we merely substitute in. Let's work with each function separately, then put them together, being sure to keep straight that 4 was given to us for the f function and 2 was given to us for the g function:

f (4) = 3(4) + 4 = 12 + 4 = 16
g (2) = 5(2) - 7 = 10 - 7 = 3

Taking the original equation and then substituting in these values, we have:

f (4) - g (2) = ?
16 - 3 = ?

and of course that equals 13.

We may also find cases where functions are nested within parentheses:

if f (x) = 3x + 4, and g(x) = 5x - 7
f (g (10)) = ?

Note that I've defined a new function for g. These are solved in the same way: by following the usual rules of math. g (10) is found inside a set of parentheses, so we start with that:

g (10) = 5(10) - 7 = 50 - 7 = 43

We then substitute this value in for g (10):

f (g (10)) = f (43)

And then we solve f (43):

f (43) = 3(43) + 4 = 129 + 4 = 134

PS: If you had to reach for your calculator to do any of that math, then you're relying on your calculator too much.