Review of Function Notation
Before we go into evaluating a function for a given value, let's first review the basics of function notation. Recall that the notation f(x) is read, "f of x". The meaning behind this notation is that we are evaluating the function for some x value. We often times refer to x as the argument for the function f.
Keep in mind that f(x) does not mean f multiplied by x. This is an easy mistake to make.
Now that we have reviewed function notation, let's take a look at what it means to evaluate a function for some value of x. Suppose we have the equation f(x) = 2x + 4 and are asked to evaluate this function when x = 5. What steps should we take?
First we need to realize that when we are being asked to evaluate a function, what we are really being asked to do is determine the function's value when the variable that it is dependent upon is equal to a certain value. In this case, the function f(x) is dependent upon x, and we want to know the value of f(5) or f(x=5). To evaluate the function for when x = 5, we simply replace each x variable with 5, as shown below:
Finding f(a) on a Graph
Sometimes you may be given the graph of a function and asked to determine some value of the function f(a). When you are asked to to this, the process is very similar to if you were given the graph of the equation and asked to find the value of y when x = 1. In this case, you would simply locate the point on the coordinate plane where x = 1 and then move vertically up or down until you touch a point on the graph for the line . Next, you move horizontally towards the y-axis and identify the point where you touch the y-axis. The y-value of the coordinate system at this point represents the y in this equation.
Now we can rewrite the previous equation with f(x) instead of y to get . If we are asked to find the value of the function when x = 1, we follow the same process we did before when x = 1. The only difference is that this time we are saying that f(1) is equal to some value instead of y. We would write f(1) = 5
Sometimes we may be asked to evaluate a function for some value given in the form f(x + a). The process we follow is the same as we did for a single argument, only now instead of replacing the x's with a's we replace them with x + a. Let's look at an example.
Suppose we are given the function f(x) = 5x - 2 and asked to evaluate the function for f(x + 2). How would we evaluate this function?
Whenever evaluating a function for some value, simply replace the variable of the function by the quantity you want to evaluate it for and then simplify.