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

Verified step by step guidance

1. Understand that the notation \((g \circ f)(2)\) means you first apply the function \(f\) to 2, then apply the function \(g\) to the result of \(f(2)\).
2. Calculate \(f(2)\) using the function \(f(x) = x + 1\). Substitute \(x\) with 2 to get \(f(2) = 2 + 1\).
3. Simplify the expression for \(f(2)\) to find the intermediate value.
4. Now, apply the function \(g\) to the result of \(f(2)\). Since \(g(x) = x^2\), substitute \(x\) with the value found in the previous step to get \(g(f(2)) = (f(2))^2\).
5. Simplify the expression \((f(2))^2\) to find the final value of \((g \circ f)(2)\).

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (g∘ƒ)(x) = g(ƒ(x)). It means you first evaluate ƒ at x, then use that output as the input for g. Understanding this process is essential to correctly evaluate composite functions.

Evaluating Functions

Evaluating a function means substituting a specific value for the variable and simplifying the expression. For example, if ƒ(x) = x + 1, then ƒ(2) = 2 + 1 = 3. This skill is necessary to find the intermediate value when composing functions.

Exponents and Squaring

Squaring a number means multiplying it by itself, as in x^2 = x × x. Recognizing how to handle exponents is important when evaluating functions like g(x) = x^2, especially after substituting values from another function.