Let f(x) = -3x + 4 and g(x) = -x² + 4x + 1. Find each of the following. Simplify if necessary. See Example 6.ƒ(0)

Verified step by step guidance

1. Identify the function f(x) = -3x + 4.
2. Substitute x = 0 into the function f(x).
3. Calculate f(0) by evaluating -3(0) + 4.
4. Simplify the expression to find the value of f(0).
5. Conclude with the simplified result of f(0).

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. For example, to find f(0) for the function f(x) = -3x + 4, you replace x with 0, resulting in f(0) = -3(0) + 4 = 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, typically expressed 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 downward trend, 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.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, represented as g(x) = ax² + bx + c. In the function g(x) = -x² + 4x + 1, the leading coefficient is negative, indicating that the parabola opens downward. Recognizing the characteristics of quadratic functions, such as their vertex and intercepts, is essential for solving problems involving them.