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

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 to find \(g(-x)\), which means we need to substitute \(-x\) into the function \(g(x)\) wherever \(x\) appears.
3. Replace every \(x\) in \(g(x)\) with \(-x\): \(g(-x) = -(-x)^{2} + 4(-x) + 1\).
4. Simplify each term carefully: \((-x)^{2}\) becomes \(x^{2}\) because squaring a negative number makes it positive; multiply \$4\( by \)-x\( to get \)-4x$.
5. Write the simplified expression for \(g(-x)\) as \(-x^{2} - 4x + 1\).

Key Concepts

Function Notation and Evaluation

Function notation, such as f(x) or g(x), represents a rule that assigns each input x to an output. Evaluating a function means substituting a specific value or expression for x and simplifying the result. For example, g(-x) means replacing every x in g(x) with -x.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. In this problem, g(x) is a quadratic polynomial, which means it includes an x² term. Understanding how to manipulate and simplify polynomials is essential.

Simplification of Algebraic Expressions

Simplification involves combining like terms and applying arithmetic operations to rewrite expressions in a simpler or more standard form. When substituting -x into g(x), it is important to carefully apply exponent rules and combine terms to get the simplest expression.