Dividing functions involves creating a new function by dividing one function by another. According to the provided YouTube video titled "Pre-Calculus - Dividing functions," this process is essentially a substitution problem.
Steps for Dividing Functions
Here's a breakdown of how to divide functions:
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Identify the functions: You'll have two functions, typically denoted as f(x) and g(x).
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Set up the division: The division of f(x) by g(x) is written as (f/g)(x) = f(x) / g(x).
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Substitute the function expressions: Replace f(x) and g(x) with their respective algebraic expressions. As the YouTube video mentions at [0:33], this involves substituting the function expressions.
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Simplify the resulting expression: Simplify the resulting rational expression by factoring and canceling common factors if possible.
Example
While the video excerpt doesn't provide a complete example, let's create one:
Suppose:
- f(x) = x2 - 4
- g(x) = x + 2
Then, to find (f/g)(x):
- (f/g)(x) = f(x) / g(x)
- (f/g)(x) = (x2 - 4) / (x + 2)
- Factor the numerator: x2 - 4 = (x + 2)(x - 2)
- (f/g)(x) = [(x + 2)(x - 2)] / (x + 2)
- Cancel the common factor (x + 2): (f/g)(x) = x - 2
Important Considerations
- Domain: When dividing functions, it's crucial to consider the domain of the resulting function. Specifically, you must exclude any values of x that make the denominator, g(x), equal to zero. In the example above, even though we simplified to x - 2, the original expression (x2 - 4) / (x + 2) is undefined when x = -2. Therefore, the domain of (f/g)(x) is all real numbers except x = -2.
