Learning Objectives:

In this section, you will learn:

• how to evaluate a given function

• the meaning of functions

Quick Review:

First, let’s start with a recap of the concepts. As you remember from previous videos, a factory is an apt simile for a function. One thing goes in and something new comes out.
$$Y = f(x)$$

“apt simile” feels a bit to verbose for the kiddos.

In this case $$Y$$ is the factory. And $$f(x)$$ is what happens in the factory, which we can liken to the blueprint. Of course, $$f(x)$$ right now is unspecified – there are no instructions.

I totally understand what you are saying mathematically, the simile was more that x=”factory in” f() = “factory” and y = “factory out”. If you feel uncomfortable with that simile let’s talk about it.

In math, the inputs and outputs are represented by variables (like $$x$$). But when you evaluate the function, you substitute a numerical value for the variable and get a number. Therefore, when you evaluate a function, the input is a number and the output is a number.

“…just like in our factory where we put coal in and got shoes out…” Let’s just rehash the simile.

Example:

Let’s look at an example:

“Let’s look at an example and see what really happens ‘behind the scenes’ when we replace x with a number and solve for the corresponding y…” Try something like this. Rember we always want to over-emphasize everything!

$$f(x) = x² + 3x + 3$$

Recalling our factory simile, let’s recognize that the expression $$x² + 3x + 3$$ is the blueprint in the factory – what is happening in the factory to transform the input into the output.

To evaluate this function, all you have to do is provide an input. In this case, the input will be a value of $$x$$. Let’s use $$x = 4$$, in which case, we simply replace all the $$x$$’s with $$4$$.

1. $$f(4) = (4)^2 + 3(4) + 3$$
2. $$f(4) = 16 + 12 + 3$$
3. $$f(4) = 31$$

We can now say that this function has been evaluated for a value of $$x = 4$$ !

Let’s try another example. Evaluate the following function $$g(b)$$ for $$g$$ values $$0$$ and $$3$$.

$$g(b) = g^3 + 2g – 14$$

Think we got our variables mixed up here. thinking these Gs should be Bs.

Don’t be scared by the notation $$g(b)$$ ! A function can be named anything and have any variables.

Do as we did above and simply sub in the values of $$b$$.

“…Try evaluating the fuctions on your own! First replace…  Then replace… Then check your answers by hovering over the practice cards to check your work!”

$$g(b) = g^3 + 2g - 14$$

Evaluate the function for the value $$b = 0$$

ANSWER: $$g(0) = - 14$$

1. $$b =0$$
2. $$g(0) = (0)^3 + 2(0) - 14$$
3. $$g(0) = - 14$$

$$g(b) = g^3 + 2g - 14$$

Evaluate the function for the value $$b = 0$$

ANSWER: $$g(0) = - 14$$

1. $$b =3$$
2. $$g(3) = (3)^3 + 2(3) - 14$$
3. $$g(3) = 1$$
You can now say that you have evaluated this function for value $$b = 0, 3$$!