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

Verified step by step guidance

1. Identify the given functions: \(f(x) = -3x + 4\) and \(g(x) = -x^2 + 4x + 1\).
2. To find \(f(a+4)\), substitute the expression \(a+4\) in place of \(x\) in the function \(f(x)\).
3. Write the substitution explicitly: \(f(a+4) = -3(a+4) + 4\).
4. Apply the distributive property to multiply \(-3\) by each term inside the parentheses: \(-3 \times a\) and \(-3 \times 4\).
5. Simplify the expression by combining like terms to get the final simplified form of \(f(a+4)\).

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.

Substitution in Functions

Substitution involves replacing the variable in a function with a given expression, like a+4. This requires careful algebraic manipulation to simplify the resulting expression correctly.

Simplifying Algebraic Expressions

Simplifying expressions means combining like terms and performing arithmetic operations to write the expression in its simplest form. This step ensures the final answer is clear and concise.