Find f/g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15

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Step 1: Write the expression for the division of the two functions, f(x) and g(x). This is represented as (f/g)(x) = f(x) / g(x). Substituting the given functions, we have (f/g)(x) = (3 - x²) / (x² + 2x - 18).

Step 2: Factorize the denominator g(x) = x² + 2x - 18. To do this, find two numbers that multiply to -18 and add to 2. These numbers are 6 and -3, so g(x) = (x + 6)(x - 3).

Step 3: Rewrite the expression for (f/g)(x) using the factored form of g(x). This gives (f/g)(x) = (3 - x²) / [(x + 6)(x - 3)].

Step 4: Determine the domain of the function. The domain is all real numbers except where the denominator equals zero. Set (x + 6)(x - 3) = 0 and solve for x. This gives x = -6 and x = 3. These values are excluded from the domain.

Step 5: Conclude that the domain of (f/g)(x) is all real numbers except x = -6 and x = 3. Express this in interval notation as (-∞, -6) ∪ (-6, 3) ∪ (3, ∞).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Division

Function division involves creating a new function by dividing one function by another. In this case, f/g means taking the function f(x) and dividing it by g(x). This operation requires understanding how to manipulate algebraic expressions and the implications of division, particularly when the denominator can affect the overall function.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by the values that make the denominator zero, as division by zero is undefined. Therefore, determining the domain involves finding the roots of the denominator and excluding them from the set of real numbers.

Factoring Quadratic Expressions

Factoring quadratic expressions is a method used to simplify polynomials, particularly those in the form ax² + bx + c. This process involves rewriting the quadratic as a product of two binomials, which can help identify the roots of the equation. In the context of the given functions, factoring g(x) will be essential to find the values that restrict the domain.

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