Let f(x) = -3x + 4 and g(x) = -x² + 4x + 1. Find each of the following. Simplify if necessary. See Example 6. ƒ(⅓)

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Identify the function f(x) given as \(f(x) = -3x + 4\).

Substitute the value \(x = \frac{1}{3}\) into the function \(f(x)\) to find \(f\left(\frac{1}{3}\right)\).

Write the substitution explicitly: \(f\left(\frac{1}{3}\right) = -3 \times \frac{1}{3} + 4\).

Simplify the multiplication part: \(-3 \times \frac{1}{3} = -1\).

Add the results to get the simplified expression: \(-1 + 4\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 ƒ(⅓), replace x with ⅓ in the expression for ƒ(x) and simplify the result.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers with coefficients. Understanding how to manipulate and simplify polynomials, such as -x² + 4x + 1, is essential for evaluating functions like 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 crucial when evaluating functions to ensure the final answer is clear and concise.

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