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

Verified step by step guidance

1. Identify the given functions: \(f(x) = -3x + 4\) and \(g(x) = -x^{2} + 4x + 1\).
2. The problem asks for \(f(x+2)\), which means we need to evaluate the function \(f\) at the input \((x+2)\) instead of \(x\).
3. Substitute \((x+2)\) into the function \(f(x)\) wherever you see \(x\). So, replace \(x\) with \((x+2)\) in the expression \(-3x + 4\).
4. Write the expression after substitution: \(f(x+2) = -3(x+2) + 4\).
5. Simplify the expression by distributing \(-3\) and combining like terms: \(f(x+2) = -3x - 6 + 4\).

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 input means substituting that input into the function's formula and simplifying the result.

Function Composition and Input Substitution

When given an expression like ƒ(x+2), the input to the function ƒ is the entire expression (x+2). This requires substituting (x+2) wherever x appears in ƒ(x) and then simplifying the resulting expression.

Simplifying Algebraic Expressions

After substitution, simplifying involves applying algebraic operations such as distribution, combining like terms, and reducing the expression to its simplest form to clearly express the function's output.