11th Grade Q1

Dividing two functions is like finding the ratio of their outputs at the same input. Here's how you do it: To divide $f(x)$ by $g(x)$, you combine their expressions as:

Example

1. Start with the functions: Write the two functions you want to divide. For example:

1. Set up the division: Substitute the expressions for $f(x)$ and $g(x)$:

1. Simplify (if possible): Factorize the denominator, if needed. In this case:

So the expression becomes:

There is no further simplification here since the numerator and denominator have no common factors.

1. Interpret the result: The new function $\frac{f}{g}$ represents the division of the two original functions. You can evaluate it for any $x$ within the domain (where $g(x)\ne 0$). For example, if $x=2$, then:

Important Notes on Domain

• The function $g(x)$ must not be zero, as division by zero is undefined.
• In this example, $g(x)=x^2-1$ is zero when $x=1$ or $x=-1$, so these values are excluded from the domain.

Summary

• Write the division $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$.
• Simplify the expression by factoring if possible.
• Exclude values of $x$ that make $g(x)=0$ from the domain.
• Use the result to evaluate the function for any input within the domain.