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

Verified step by step guidance

1. Identify the function g(x) given as \(g(x) = -x^2 + 4x + 1\).
2. To find \(g(k)\), substitute the variable \(x\) in the function \(g(x)\) with \(k\).
3. Write the expression after substitution: \(g(k) = -(k)^2 + 4(k) + 1\).
4. Simplify the expression by squaring \(k\) and distributing the negative sign: \(g(k) = -k^2 + 4k + 1\).
5. The simplified expression \(g(k) = -k^2 + 4k + 1\) is the value of the function \(g\) at \(k\).

Key Concepts

Function Notation and Evaluation

Function notation, such as g(k), represents the output of a function g when the input is k. To evaluate g(k), substitute the value k into the function's formula and simplify the expression to find the result.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding how to work with polynomials, like -x^2 + 4x + 1, is essential for evaluating and simplifying function values.

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.