Let f(x) = -3x + 4 and g(x) = -x² + 4x + 1. Find each of the following. Simplify if necessary. See Example 6. ƒ(x + 2)

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Identify the given functions: \(f(x) = -3x + 4\) and \(g(x) = -x^{2} + 4x + 1\).

To find \(f(x + 2)\), substitute every occurrence of \(x\) in \(f(x)\) with \((x + 2)\).

Write the substitution explicitly: \(f(x + 2) = -3(x + 2) + 4\).

Apply the distributive property to expand: \(-3(x + 2) = -3x - 6\).

Combine like terms to simplify: \(f(x + 2) = -3x - 6 + 4 = -3x - 2\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Notation and Evaluation

Function notation, such as f(x), represents a rule that assigns each input x to an output. Evaluating f(x + 2) means substituting x + 2 into the function in place of x, then simplifying the expression to find the output.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers with coefficients. Understanding how to manipulate and simplify polynomials, such as linear (f(x) = -3x + 4) and quadratic (g(x) = -x² + 4x + 1), is essential for evaluating and combining functions.

Simplification of Algebraic Expressions

Simplification involves combining like terms and performing arithmetic operations to write expressions in their simplest form. After substituting values into functions, simplifying ensures the final answer is clear and concise.

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