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

Verified step by step guidance

1. Understand that the notation \((\f+\g)(2)\) means you need to find the value of the function \((\f+\g)\) at \(x=2\). The function \((\f+\g)(x)\) is defined as \(\f(x) + \g(x)\).
2. Write the expression for \((\f+\g)(x)\) by adding the two given functions: \(\f(x) = x + 1\) and \(\g(x) = x^2\). So, \((\f+\g)(x) = (x + 1) + x^2\).
3. Substitute \(x = 2\) into the expression: \((\f+\g)(2) = (2 + 1) + 2^2\).
4. Simplify the terms inside the parentheses and the exponent: \((2 + 1) = 3\) and \$2^2 = 4$.
5. Add the simplified values together to get the final result: \$3 + 4$.

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 Addition (Sum of Functions)

The sum of two functions (ƒ + g)(x) is defined as ƒ(x) + g(x). To evaluate (ƒ + g)(2), you find ƒ(2) and g(2) separately, then add the results together.

Substitution and Simplification

Substitution involves replacing the variable x with a given number in the function's expression. Simplification means performing arithmetic operations to reduce the expression to a single numerical value.