Without using paper and pencil, evaluate each expression given the following functions. $\text{ƒ}\left(x\right)=x+1$ and $g\left(x\right)=x^2$
$\left(\text{ƒ}/g\right)\left(2\right)$

Verified step by step guidance

1. Understand that the notation (ƒ/g)(2) means the function ƒ divided by the function g, evaluated at x = 2. In other words, (ƒ/g)(2) = \(\frac{ƒ(2)}{g(2)}\).
2. Evaluate ƒ(2) using the function definition ƒ(x) = x + 1. Substitute x = 2 to get ƒ(2) = 2 + 1.
3. Evaluate g(2) using the function definition g(x) = x^2. Substitute x = 2 to get g(2) = 2^2.
4. Write the expression for (ƒ/g)(2) as \(\frac{ƒ(2)}{g(2)}\) = \(\frac{2 + 1}{2^2}\).
5. Simplify the fraction by performing the arithmetic operations in the numerator and denominator separately, then write the final simplified fraction.

Key Concepts

Function Notation and Evaluation

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

Function Operations (Division of Functions)

The division of two functions (ƒ/g)(x) means creating a new function by dividing the output of ƒ(x) by the output of g(x), expressed as (ƒ/g)(x) = ƒ(x) / g(x). It requires evaluating both functions at the same input and then dividing the results.

Substitution and Simplification

To evaluate (ƒ/g)(2), substitute 2 into both ƒ(x) and g(x), then divide the results. Simplifying the expression involves performing arithmetic operations carefully to find the final value without using paper and pencil.