Mastering how to multiply functions might feel a little scary at first, but don’t worry! With the right tips and practice, it can become an easy part of your math skills in Algebra II. Here are some helpful secrets to make you a pro at this important math topic.

What is a Function?

First, let's break down what we mean by a function.

A function is like a machine that takes an input and gives one specific output for that input. When we multiply functions, we make a new function, which we can write as $(f \cdot g)(x)$. Here, $f(x)$ and $g(x)$ are our original functions, and $(f \cdot g)(x)$ is their product.

The Basics of Multiplying Functions

Multiplying two functions is pretty simple. You just multiply their outputs. For example, if we have these two functions:

• $f(x) = 2x + 3$
• $g(x) = x^2 - 1$

We can multiply them and write it like this:

$(f \cdot g)(x) = f(x) \cdot g(x) = (2x + 3)(x^2 - 1)$

Step-by-Step Guide

Here’s how you can do it step by step:

1. Substitute Values: First, plug in the input $x$ into both functions.

2. Multiply the Results: Then, multiply the results from the two functions.

Let’s use our earlier example to see this in action:

• For $x = 2$:
  • $f(2) = 2(2) + 3 = 7$
  • $g(2) = (2)^2 - 1 = 3$
  • Now, we find $(f \cdot g)(2) = 7 \cdot 3 = 21$

Expanding the Product

You might also need to expand the expression $(f \cdot g)(x)$ into a polynomial. So, using our example:

$(f \cdot g)(x) = (2x + 3)(x^2 - 1)$

To expand this, we use something called the distributive property (you might know it as FOIL):

$= 2x \cdot x^2 + 2x \cdot (-1) + 3 \cdot x^2 + 3 \cdot (-1)$ $= 2x^3 - 2x + 3x^2 - 3$

So, the product function becomes:

$(f \cdot g)(x) = 2x^3 + 3x^2 - 2x - 3$

Practice Makes Perfect

Finally, the best way to get good at multiplying functions is to practice! Try different functions, double-check your work, and even attempt harder examples with quadratic and cubic functions. The more you practice, the easier it will become, and you’ll understand better how function multiplication works in the big picture of Algebra II.

Happy studying!