Let ƒ(x)=-3x+4 and g(x)=-x^2+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) which is given as $g(x)=-x^2+4x+1$.
2. Substitute $x=\frac{1}{2}$ into the function g(x).
3. Calculate $g\left(\frac{1}{2}\right)=-{\left(\frac{1}{2}\right)}^2+4\left(\frac{1}{2}\right)+1$.
4. Simplify $-{\left(\frac{1}{2}\right)}^2$ to $-\frac{1}{4}$.
5. Combine all terms: $-\frac{1}{4}+2+1$.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. For example, to find g(1/2), you replace x in the function g(x) with 1/2, allowing you to calculate the corresponding output value. This process is fundamental in understanding how functions operate and how to manipulate them.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax^2 + bx + c. In this case, g(x) = -x^2 + 4x + 1 is a quadratic function where the leading coefficient is negative, indicating that the parabola opens downwards. Understanding the properties of quadratic functions, such as their vertex and intercepts, is essential for analyzing their behavior.

Simplification of Expressions

Simplification involves reducing mathematical expressions to their simplest form, making them easier to work with. This can include combining like terms, factoring, or performing arithmetic operations. In the context of evaluating g(1/2), simplifying the resulting expression helps clarify the final output and ensures accuracy in calculations.