For the pair of functions defined, find (ƒ/g)(x). Give the domain of each. See Example 2.
ƒ(x)=2x^2-3x, g(x)=x^2-x+3

Verified step by step guidance

1. First, recall that the operation (ƒ/g)(x) means the quotient of the two functions, which is given by the formula: \[(\frac{ƒ}{g})(x) = \frac{ƒ(x)}{g(x)}\]
2. Substitute the given functions into the quotient: \[(\frac{ƒ}{g})(x) = \frac{2x^2 - 3x}{x^2 - x + 3}\]
3. Next, simplify the expression if possible by factoring the numerator and denominator and canceling any common factors. For the numerator, factor out the common term: \[2x^2 - 3x = x(2x - 3)\]. Check if the denominator can be factored as well.
4. Determine the domain of each function separately. For \(ƒ(x) = 2x^2 - 3x\), since it is a polynomial, its domain is all real numbers: \[(-\infty, \infty)\].
5. For \(g(x) = x^2 - x + 3\), also a polynomial, the domain is all real numbers. However, for the quotient \((ƒ/g)(x)\), the domain excludes values where \(g(x) = 0\) because division by zero is undefined. Solve the equation \[x^2 - x + 3 = 0\] to find any such values and exclude them from the domain.

Key Concepts

Function Operations (Division of Functions)

Dividing two functions ƒ and g, denoted (ƒ/g)(x), means creating a new function by dividing the output of ƒ(x) by g(x). This operation is defined as (ƒ/g)(x) = ƒ(x) / g(x), where g(x) ≠ 0 to avoid division by zero.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. For (ƒ/g)(x), the domain includes all x-values in the domains of ƒ and g, except where g(x) = 0, since division by zero is undefined.

Polynomial Functions and Their Domains

Both ƒ(x) = 2x² - 3x and g(x) = x² - x + 3 are polynomial functions, which are defined for all real numbers. However, when dividing, the domain excludes values that make the denominator zero, so identifying zeros of g(x) is essential.