91Ó°ÊÓ

Function addition is the process of creating a new function by adding two existing functions together. For two functions \( f(x) \) and \( g(x) \), the sum, represented as \( (f+g)(x) \), is obtained by simply adding the outputs of \( f \) and \( g \) for each input \( x \). For example, if \( f(x) = x - 8 \) and \( g(x) = 5x^2 \), then:

• \( (f+g)(x) = f(x) + g(x) = (x - 8) + 5x^2 = 5x^2 + x - 8 \)

The new function \( 5x^2 + x - 8 \) inherits the domain from \( f \) and \( g \). Since both are defined for all real numbers, \( f+g \) is also defined for all real numbers.

Function subtraction works similarly to addition, except you subtract one function from another. With \( f(x) \) and \( g(x) \), subtraction is shown as \( (f-g)(x) \). This involves taking the value of \( g(x) \) away from \( f(x) \) for each input \( x \). For the functions \( f(x) = x - 8 \) and \( g(x) = 5x^2 \):

• \( (f-g)(x) = f(x) - g(x) = (x - 8) - 5x^2 = -5x^2 + x - 8 \)

This function \( -5x^2 + x - 8 \) also has a domain of all real numbers since both subtracting functions are defined everywhere. Subtracting functions in calculus follows straightforward arithmetic rules but results in a new function with potentially different properties.

Function multiplication combines functions by multiplying their outputs for each input \( x \). The notation for this is \( (f \cdot g)(x) \). Given \( f(x) = x - 8 \) and \( g(x) = 5x^2 \), the process is:

• \( (f \cdot g)(x) = f(x) \cdot g(x) = (x - 8)(5x^2) = 5x^3 - 40x^2 \)

The resulting function \( 5x^3 - 40x^2 \) is a polynomial. Polynomials have simple domain characteristics—they're defined for all real numbers. Multiplying functions can produce changes in degree and nature of the function, although it usually maintains the same domain.

Function division involves dividing the output of one function by the output of another and is represented as \( \left( \frac{f}{g} \right)(x) \). This operation introduces more complexity because one must be wary of dividing by zero. For \( f(x) = x - 8 \) and \( g(x) = 5x^2 \):

• \( \left( \frac{f}{g} \right)(x) = \frac{x - 8}{5x^2} \)

The domain is all real numbers except where \( g(x) = 0 \). In this case, \( g(x) \) is zero when \( x = 0 \), creating a restriction in the domain. Therefore, the function \( \frac{x - 8}{5x^2} \) is defined everywhere except \( x = 0 \). This creates a hole in the graph at \( x = 0 \), crucial when analyzing the behavior of divided functions.

The domain of a function refers to all possible inputs (or \( x \) values) for which the function is defined. It can vary depending on the operations involved. To determine the domain, consider situations in which the function fails to produce a valid output.

• For addition, subtraction, and multiplication of polynomials, the domain is all real numbers.
• When division is involved, the domain is restricted by the values that make the denominator zero, which must be excluded from the domain.

In our example, the domain for the addition, subtraction, and multiplication functions extends over all real numbers because both \( f(x) = x - 8 \) and \( g(x) = 5x^2 \) are polynomials, which are always defined.
The function \( \left( \frac{f}{g} \right)(x) \) has a domain of all real numbers except where \( g(x) = 0 \). Here, \( \left( \frac{f}{g} \right)(x) \) is undefined at \( x = 0 \). Thus, it's crucial to evaluate functions before assuming they are well-defined across all real numbers.