Without using paper and pencil, evaluate each expression given the following functions. $ƒ (x) = x + 1$ and $g (x) = x^{2}$
$open paren g composed with f with hook close paren times 2$

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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 $f(2)$.

Calculate $f(2)$ using the function $f(x) = x + 1$. Substitute $x$ with 2 to get $f(2) = 2 + 1$.

Simplify the expression for $f(2)$ to find the intermediate value.

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 $g(f(2)) = (f(2))^2$.

Simplify the expression $(f(2))^2$ to find the final value of $(g \circ f)(2)$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1ƒ(x)=x+1 and g(x)=x2g(x)=x^2 (g∘ƒ)(2)(g∘ƒ)(2)