Let ƒ(x)=-3x+4 and g(x)=-x²+4x+1. Find each of the following. Simplify if necessary. See Example 6. 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 \(g(x)\), so write \(g\left(\frac{1}{2}\right) = -\left(\frac{1}{2}\right)^2 + 4\left(\frac{1}{2}\right) + 1\).
3. Calculate each term separately: first, square \(\frac{1}{2}\) to get \(\left(\frac{1}{2}\right)^2\), then multiply by \(-1\); next, multiply \$4\( by \)\frac{1}{2}\(; finally, keep the constant term \)1$ as is.
4. Combine all the terms by adding and subtracting them according to the expression \(-\left(\frac{1}{2}\right)^2 + 4\left(\frac{1}{2}\right) + 1\).
5. Simplify the resulting expression to get the final value of \(g\left(\frac{1}{2}\right)\).

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.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding how to handle terms like -x^2 and 4x is essential for correctly evaluating and simplifying the function.

Simplification of Algebraic Expressions

Simplification involves performing arithmetic operations and combining like terms to write expressions in their simplest form. This step ensures the final answer is clear and concise, especially after substituting values into polynomial functions.