In the study of functions, understanding how to multiply and divide them is essential, as it allows for the exploration of new relationships and behaviors between different mathematical expressions. When multiplying two functions, such as \( f(x) = \sqrt{x} \) and \( g(x) = 3x - 6 \), the product is obtained by distributing one function into the other. This results in \( f \cdot g = \sqrt{x} (3x - 6) = 3x\sqrt{x} - 6\sqrt{x} \). The domain of the resulting function is determined by the intersection of the individual domains of \( f \) and \( g \). For \( f(x) \), the domain is \( [0, \infty) \) since the square root function is defined for non-negative values. For \( g(x) \), which is a polynomial, the domain is all real numbers, \( (-\infty, \infty) \). Therefore, the domain of the product \( f \cdot g \) is \( [0, \infty) \), as it is the more restrictive of the two domains.
When dividing functions, the process is similar but requires additional caution regarding the denominator. For the same functions, the division is expressed as \( \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{3x - 6} \). Here, the domain of \( f(x) \) remains \( [0, \infty) \), but \( g(x) \) introduces a restriction: the denominator cannot equal zero. Setting \( 3x - 6 = 0 \) gives \( x = 2 \), which must be excluded from the domain. Thus, the domain of the quotient is \( [0, 2) \cup (2, \infty) \), allowing all positive values except for 2.
In another example, consider \( f(x) = x^2 - 4 \) and \( g(x) = x + 2 \). The product \( f \cdot g \) can be calculated using the FOIL method, yielding \( x^3 + 2x^2 - 4x - 8 \). Since both functions are polynomials, their combined domain is all real numbers, \( (-\infty, \infty) \). However, when dividing \( f(x) \) by \( g(x) \), the restriction arises from the denominator \( x + 2 \), which cannot be zero. This leads to the exclusion of \( x = -2 \) from the domain, resulting in \( (-\infty, -2) \cup (-2, \infty) \) for the quotient.
In summary, when performing operations on functions, it is crucial to consider the domains carefully, especially when division is involved, as restrictions can significantly affect the resulting function's domain. Understanding these principles will enhance your ability to work with functions effectively.
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