[MUSIC PLAYING] Let's look at our objectives for today. We'll start by defining a function. We'll then talk about how to interpret function notation. And finally, we'll do some examples evaluating functions.
Let's start by defining a function. A function is a relation between two variables in which an input variable corresponds to exactly one output variable. These variables can be referred to as inputs and outputs because the value for one variable is put into a function. The function provides a specific set of operations to output a specific value for another variable.
Here is an example. Suppose we have the rule-- take the input, multiply by 2, and add 3. If we use the input 5 with this rule, the output would be 13 because 5 times 2 is 10 and 10 plus 3 is 13. If we use the input 8 with this rule, the output would be 19 because 8 times 2 is 16 and 16 plus 3 is 19. And if we use the input negative 10, the output would be negative 17 because negative 10 times 2 is negative 20 and negative 20 plus 3 is negative 17.
We commonly use f (x) for function notation. It is read f of x and does not mean f multiplied by x. Function notation is used to name a function where x is the independent variable or the input. f of x then is used to represent the dependent variable or output of the function so it is the same as the variable y. Another example of function notation would be g of t, where t is the independent variable or the input of the function g of t.
Now, let's do some examples evaluating functions. Here's an example of a function. f of x equals 4x squared minus 6. To evaluate a function means to find the value of the output f of x for a given input x. So to evaluate a function, we replace each x in the function with the input value and use order of operations to simplify the expression to determine the output value.
For example, we can evaluate the function f of x for x equals 10, which means we replace each x with 10. So we have f of 10 equals 4 times 10 squared minus 6. We start with our exponent, so we have 4 times 100 minus 6.
We then multiply 4 times 100, which is 400, and then we subtract 6. 400 minus 6 will be 394, so f of 10 is 394. Remember, we read this as f of 10, and it does not mean to multiply f and 10 together. We can also represent our solution as an ordered pair-- 10, 394, which would be on the graph of this function.
Let's also evaluate the function f of x for x equals negative 4. So now, we will replace each x with negative 4. So we have f of negative 4 equals 4 times negative 4 squared minus 6. We, again, start with our exponent. Negative 4 squared is a positive 16, so we have 4 times 16 minus 6.
We then multiply. 4 times 16 is 64. And finally, we subtract 6. 64 minus 6 is 58, so f of negative 4 is 58. Again, we can write this as an ordered pair-- negative 4, 58, which is a point that would be on the graph of the function f of x.
Let's look at our important points for today. Make sure you get these in your notes so you can refer to them later. A function is a relation between two variables in which an input variable corresponds to exactly one output variable.
These variables can be referred to as inputs and outputs because the value for one variable is put into a function to output a specific value for another variable. Function notation is used to name a function where x is the independent variable or the input of the function. And f of x is used to represent the dependent variable or output of the function, so it is the same as the variable y.
So I hope these important points and examples helped you understand a little bit more about evaluating a function. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
