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

Verified step by step guidance

1. Identify the function f(x) = -3x + 4.
2. Substitute x = \frac{1}{3} into the function f(x).
3. Calculate f(\frac{1}{3}) by replacing x with \frac{1}{3} in the expression: f(\frac{1}{3}) = -3(\frac{1}{3}) + 4.
4. Simplify the expression: -3(\frac{1}{3}) + 4.
5. Combine the terms to find the simplified result.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, to evaluate f(x) at x = ⅓, you replace x in the function f(x) = -3x + 4 with ⅓, resulting in f(⅓) = -3(⅓) + 4. This process is fundamental in understanding how functions behave at particular points.

Linear Functions

A linear function is a polynomial function of degree one, represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the given function f(x) = -3x + 4, the slope is -3, indicating a decrease in value as x increases, while the y-intercept is 4, showing where the line crosses the y-axis. Understanding linear functions is crucial for analyzing their graphs and behaviors.

Simplification of Expressions

Simplification of expressions 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 f(⅓), simplifying the result after substitution ensures clarity and accuracy in the final answer.