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

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 \(f(p)\), which means we need to evaluate the function \(f\) at the input \(p\).
3. To find \(f(p)\), substitute \(p\) in place of \(x\) in the function \(f(x)\).
4. Write the expression for \(f(p)\) as \(f(p) = -3p + 4\).
5. Since this expression is already simplified, this is the final form of \(f(p)\).

Key Concepts

Function Notation and Evaluation

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

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding how to work with linear (degree 1) and quadratic (degree 2) polynomials is essential for evaluating and simplifying expressions like ƒ(p) and g(x).

Simplification of Algebraic Expressions

Simplification involves combining like terms and performing arithmetic operations to write expressions in their simplest form. This skill is necessary after substituting values into functions to present the final answer clearly and concisely.