91Ó°ÊÓ

Linear functions are fundamental in mathematics and are characterized by their constant rate of change. A linear function can be represented in the form of \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The slope \( m \) indicates how steep the line is and in which direction it moves. If \( m \) is positive, the function increases; if negative, the function decreases. Here, in the function \( f(x) = 4 - 5x \), it is expressed in the form of \( m \equiv -5 \), showing that for every unit increase in \( x \), \( f(x) \) decreases by 5 units. This simplicity makes linear equations easy to visualize as straight lines on a graph, allowing us to predict their behavior over different intervals.

The process of function evaluation involves finding the output of a function for specific inputs. To evaluate a function like \( f(x) = 4 - 5x \) at a particular value, substitute the value of \( x \) into the function and simplify.

• At \( x = 3 \), the function evaluates to \( f(3) = 4 - 5 imes 3 = -11 \).
• At \( x = 5 \), it evaluates to \( f(5) = 4 - 5 imes 5 = -21 \).

By plugging these values into the function, you can easily determine what the function equals for particular inputs. This can form the basis for analyzing how the function behaves over different values, such as determining the net change.

Algebra is the branch of mathematics that helps in manipulating equations to find unknown values. In this context, it plays a crucial role in calculating the net change of a function and solving problems involving linear equations. Algebra allows us to replace variables with numbers, simplifying expressions and solving them step-by-step. Here, to find the net change from \( x = 3 \) to \( x = 5 \), you compute:

• First, determine \( f(3) = -11 \).
• Next, find \( f(5) = -21 \).
• Finally, calculate the net change: \( f(5) - f(3) = -21 - (-11) = -10 \).

This process shows the power of algebra in systematically determining how values change, thus providing insights into the behavior of functional relationships over specified intervals.