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

Verified step by step guidance

1. Identify the function \( g(x) = -x^2 + 4x + 1 \). We need to find \( g(-x) \), which means we substitute \( -x \) wherever we see \( x \) in the function \( g(x) \).
2. Replace every \( x \) in the expression for \( g(x) \) with \( -x \). This gives us \( g(-x) = -(-x)^2 + 4(-x) + 1 \).
3. Simplify the expression step-by-step: first, calculate \( (-x)^2 \), then multiply by \( -1 \), next multiply \( 4 \) by \( -x \), and finally add \( 1 \).
4. Write the simplified expression clearly, combining like terms if possible to get the final form of \( g(-x) \).
5. Double-check your simplification to ensure all signs and powers are correctly handled.

Key Concepts

Function Notation and Evaluation

Function notation, such as g(x), represents a rule that assigns each input x to an output. Evaluating a function at a specific value means substituting that value into the function's formula and simplifying the result.

Substitution of Expressions into Functions

Substitution involves replacing the variable in a function with another expression, like -x in g(-x). This requires careful algebraic manipulation to correctly simplify the resulting expression.

Simplifying Polynomial Expressions

Simplifying polynomials involves combining like terms and applying exponent rules. When substituting expressions, it is important to expand powers and distribute coefficients properly to write the function in its simplest form.