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. Understand that the notation (ƒ/g)(5) means the value of the function ƒ(x) divided by g(x) evaluated at x = 5, which can be written as \( \frac{ƒ(5)}{g(5)} \).
2. Calculate \( ƒ(5) \) by substituting 5 into the function ƒ(x) = \( x^2 + 3 \). This gives \( ƒ(5) = 5^2 + 3 \).
3. Calculate \( g(5) \) by substituting 5 into the function g(x) = \( -2x + 6 \). This gives \( g(5) = -2(5) + 6 \).
4. Form the quotient \( \frac{ƒ(5)}{g(5)} \) by dividing the result from step 2 by the result from step 3.
5. Simplify the expression from step 4 to find the final value of \( (ƒ/g)(5) \).

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 division of two functions (ƒ/g)(x) is defined as the quotient of their outputs: ƒ(x) divided by g(x). To find (ƒ/g)(5), evaluate both ƒ(5) and g(5) separately, then divide the results, ensuring the denominator is not zero.

Polynomial and Linear Functions

ƒ(x) = x² + 3 is a polynomial function of degree 2, and g(x) = -2x + 6 is a linear function. Understanding their forms helps in correctly substituting values and simplifying expressions during evaluation.