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

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Understand that the expression (ƒ - g)(2) means you need to find the value of the function (ƒ - g) at x = 2. The function (ƒ - g)(x) is defined as ƒ(x) - g(x).

Write the expression for (ƒ - g)(x) by subtracting g(x) from ƒ(x): (ƒ - g)(x) = ƒ(x) - g(x) = (x + 1) - (x^2).

Substitute x = 2 into the expression: (ƒ - g)(2) = (2 + 1) - (2^2).

Simplify each part separately: calculate 2 + 1 and 2^2.

Subtract the results from the previous step to find the value of (ƒ - g)(2).

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주요 개념

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

Function Notation and Evaluation

Function notation, such as ƒ(x) and g(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 (Difference of Functions)

The difference of two functions (ƒ - g)(x) is defined as ƒ(x) minus g(x). To evaluate (ƒ - g)(2), you find ƒ(2) and g(2) separately, then subtract g(2) from ƒ(2). This operation combines functions to create a new function.

Substitution and Simplification

Substitution involves replacing the variable x with a given number in the function's expression. After substitution, simplifying the resulting expression by performing arithmetic operations yields the final value of the function at that point.

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 (ƒ−g)(2)(ƒ-g)(2)