Multiplication and division are important ways to combine functions. They can change how the functions behave. In Algebra II, knowing how to use these operations with functions is really helpful for solving tougher problems.

Multiplication of Functions

When we multiply two functions, like $f(x)$ and $g(x)$, we get a new function:

$$h(x) = f(x) \cdot g(x).$$

This multiplication usually changes the height and shape of the graphs. For example, if we have $f(x) = x$ and $g(x) = x^2$, then:

$$h(x) = x \cdot x^2 = x^3.$$

Here, the new function $h(x)$ gets steeper as $x$ increases. If one of the functions is a constant number (like $k$), then multiplying by $k$ will change how fast the graph grows but won’t change the input values.

Division of Functions

When we divide functions, we write it like this:

$$h(x) = \frac{f(x)}{g(x)}.$$

This operation can bring some interesting features, like new lines where the function doesn’t work (called asymptotes) and breaks in the graph. For example, if $f(x) = x^2$ and $g(x) = x - 1$, we get:

$$h(x) = \frac{x^2}{x - 1}.$$

In this case, $h(x)$ has a vertical asymptote at $x = 1$. That means the function goes up to infinity or down to negative infinity as $x$ gets close to 1.

Key Points to Remember:

1. Multiplication Effects:

   • Can make the function values bigger or smaller based on $f(x)$ and $g(x)$.
   • Does not usually create asymptotes unless one of the functions gets close to zero.

2. Division Effects:

   • Can create vertical asymptotes where $g(x)$ equals zero.
   • Changes the possible values ($x$ values) based on $g(x)$.

Conclusion

When we multiply or divide functions, it changes their graphs and how they act. Multiplication usually combines and scales their properties, while division adds more complexities like asymptotes and breaks. Knowing how these operations work is important for understanding function behavior in Algebra II.