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

Verified step by step guidance

1. Identify the function g(x) given as \(g(x) = -x^{2} + 4x + 1\).
2. Substitute the value \(x = \frac{1}{2}\) into the function \(g(x)\) to find \(g\left(\frac{1}{2}\right)\).
3. Write the expression after substitution: \(g\left(\frac{1}{2}\right) = -\left(\frac{1}{2}\right)^{2} + 4 \times \frac{1}{2} + 1\).
4. Simplify each term step-by-step: calculate \(\left(\frac{1}{2}\right)^{2}\), multiply \$4\( by \)\frac{1}{2}$, and then combine all terms.
5. Combine the simplified terms to write the final expression for \(g\left(\frac{1}{2}\right)\) in its simplest form.

Key Concepts

Function Evaluation

Function evaluation involves substituting a given input value into the function's formula and simplifying to find the output. For example, to find g(1/2), replace x with 1/2 in g(x) and simplify the expression.

Quadratic Functions

A quadratic function is a polynomial of degree two, typically written as ax² + bx + c. Understanding its structure helps in correctly substituting values and simplifying expressions involving squares and linear terms.

Simplification of Algebraic Expressions

Simplification involves performing arithmetic operations and combining like terms to write expressions in their simplest form. This is essential after substitution to present the final answer clearly and concisely.