# What Does it Mean to Evaluate a Function?

To evaluate a function means to find the output of a function for a given input.

# How to Evaluate a Function

To evaluate a function, substitute the value of the input for the variable in the function.

Question: Evaluate the function below when x = 2:

Step 1: Substitute 2 for x in the function.

Step 2: Use the order of operations to evaluate the expression.

The function evaluated at x = 2 is 5.

# How to Evaluate a Function Using a 'Number Machine'

Functions can be visualized as a 'number machine', where a number is put in one end, and another number comes out the other.

The function f(x) = 2x + 1 is represented by the number machine below:

To evaluate the function at x = 2, insert 2 into the number machine:

The function evaluated at x = 2 is 5.

# How to Evaluate a Function Using Graphical Methods

Functions can be plotted on graphs, with the input x on the x-axis and the output f(x) on the y-axis.

Take the function above, f(x) = x + 1, as an example. When this function is plotted, it is a line:

Question: Evaluate the function f(x) = 2x + 1 when x = 2.

Step 1: Find the input x = 2 on the x-axis. Draw a line (mentally or otherwise) straight up to the function from the x-axis.

Step 1: Draw a line (mentally or otherwise) straight across from the point of the function just found to the y-axis.

The function evaluated at x = 2 is 5.

# A Real Example of How to Evaluate a Function

The slider below shows a real life example of evaluating a function.
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##### Note
WHAT IS A FUNCTION?

A function is a relation between an input and an output.

The input is often denoted x and the function f(x).

Consider the function below:

• Let the input be denoted by the variable x.

• The relation the input plus one: x + 1.

• The output is the result of adding one to the input.

# A FUNCTION AS A NUMBER MACHINE

It can be helpful to visualize a function as a machine that takes an input and processes it into an output.

Consider the function f(x) = x + 1:

To evaluate the function at a certain value of the input, for example x = 1, put 1 into the number machine: