Let ƒ(x)=x²+3 and g(x)=-2x+6. Find each of the following. (ƒ/g)(-1)

Verified step by step guidance

1. Understand that the notation (ƒ/g)(x) means the function ƒ(x) divided by the function g(x), so (ƒ/g)(x) = \frac{ƒ(x)}{g(x)}.
2. Substitute the given functions into the expression: (ƒ/g)(x) = \frac{x^2 + 3}{-2x + 6}.
3. Replace x with -1 in both the numerator and the denominator: numerator = (-1)^2 + 3, denominator = -2(-1) + 6.
4. Calculate the values inside the numerator and denominator separately: numerator = 1 + 3, denominator = 2 + 6.
5. Write the final expression as a fraction using these values: (ƒ/g)(-1) = \frac{1 + 3}{2 + 6}.

Key Concepts

Function Notation and Evaluation

Function notation, such as ƒ(x), represents a rule that assigns each input x to an output value. Evaluating a function at a specific value means substituting that value into the function's expression and simplifying to find the output.

Function Division (Quotient of Functions)

The quotient of two functions (ƒ/g)(x) is defined as ƒ(x) divided by g(x), provided g(x) ≠ 0. To find (ƒ/g)(-1), evaluate both ƒ(-1) and g(-1) separately, then divide the results.

Domain Restrictions in Function Operations

When dividing functions, the domain excludes values where the denominator function equals zero, as division by zero is undefined. It's important to check g(x) ≠ 0 before evaluating (ƒ/g)(x) at a given point.