Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ/g)(5)

Verified step by step guidance

1. Step 1: Understand the notation (ƒ/g)(x), which represents the division of the function ƒ(x) by g(x).
2. Step 2: Write the expression for (ƒ/g)(x) as (ƒ(x))/(g(x)).
3. Step 3: Substitute the given functions into the expression: (ƒ(x))/(g(x)) = (x^2 + 3)/(-2x + 6).
4. Step 4: Substitute x = 5 into the expression: ((5)^2 + 3)/(-2(5) + 6).
5. Step 5: Simplify the expression by calculating the numerator and the denominator separately, then divide the results.

Key Concepts

Function Notation

Function notation is a way to represent mathematical functions. In this context, ƒ(x) and g(x) denote two different functions, where ƒ(x) = x² + 3 and g(x) = -2x + 6. Understanding how to evaluate these functions at specific values, such as x = 5, is crucial for solving the problem.

Division of Functions

The division of functions involves creating a new function that is the quotient of two existing functions. In this case, (ƒ/g)(x) represents the function formed by dividing ƒ(x) by g(x). To find (ƒ/g)(5), you must first evaluate ƒ(5) and g(5) and then compute the quotient of these two results.

Evaluating Functions

Evaluating functions means substituting a specific value for the variable in the function's expression. For example, to evaluate ƒ(5), you replace x with 5 in the expression x² + 3, resulting in ƒ(5) = 5² + 3. This process is essential for finding the values needed to compute (ƒ/g)(5).