Let f(x) = -3x + 4 and g(x) = -x² + 4x + 1. Find each of the following. Simplify if necessary. See Example 6. ƒ(x + 2)

Verified step by step guidance

1. Identify the function f(x) = -3x + 4.
2. Substitute x + 2 into the function f(x) to find f(x + 2).
3. Replace every instance of x in f(x) with (x + 2), resulting in f(x + 2) = -3(x + 2) + 4.
4. Distribute the -3 across the terms inside the parentheses: -3(x + 2) becomes -3x - 6.
5. Combine the terms: f(x + 2) = -3x - 6 + 4.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. In this case, to find ƒ(x + 2), you replace 'x' in the function f(x) = -3x + 4 with 'x + 2'. This process is essential for understanding how functions behave with different inputs.

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. The function f(x) = -3x + 4 is linear, indicating that its graph is a straight line. Understanding linear functions is crucial for analyzing their properties and transformations.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to work with. After evaluating ƒ(x + 2), you may need to combine like terms or reduce fractions. This skill is vital in algebra and trigonometry for solving equations and understanding function behavior.