91Ó°ÊÓ

A linear function is a mathematical expression that creates a straight line when plotted on a graph. These functions have a constant rate of change, which means for every increase in the input variable, the output variable changes by a constant amount. This relationship can be described by the function in the form of \( f(x) = ax + b \), where:

• \( a \) is the slope or gradient of the line, indicating how steep the line is.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

For instance, in the function \( f(x) = 3x + 5 \), the slope \( a \) is 3, and the y-intercept \( b \) is 5. This means that for every one unit increase in \( x \), the function \( f(x) \) increases by 3 units, and it intersects with the y-axis at 5.
This straightforward nature makes linear functions a common and easy-to-handle tool in mathematics, aiding in predicting and modeling real-world scenarios.

Function notation is a convenient way to express a function in mathematics. It uses the symbol \( f(x) \) to represent a function applied to a variable \( x \). This notation not only specifies the operation carried out by the function but also makes it clear which variable is considered the input.
When you see a function written like \( f(x) = 3x + 5 \), it means that for an input \( x \), you multiply it by 3 and add 5 to get the output. Function notation is particularly helpful in distinguishing between different functions and can easily adapt to express complex relationships.
One key feature of function notation is that it allows us to express the inverse of a function efficiently. For example, the inverse of \( f(x) = 3x + 5 \), written as \( f^{-1}(x) \), means we are looking for the function that "undoes" \( f(x) \), restoring the input \( x \) from a given output.

Solving an equation means finding the value of the variable that makes the equation true. When dealing with linear equations like our exercise, solving involves basic operations to isolate the variable of interest on one side of the equation.
Let's walk through the solution for finding the inverse of \( f(x) = 3x + 5 \) to illustrate this concept. Initially, we replaced \( f(x) \) with \( y \) and got the equation \( y = 3x + 5 \). After swapping the variables to start isolating \( y \), we ended with \( x = 3y + 5 \).
To isolate \( y \), subtract 5 from both sides obtaining \( x - 5 = 3y \), and then divide each side by 3 leading to \( y = \frac{x - 5}{3} \). These steps are examples of solving equations by performing equivalent operations that maintain equality, gradually transforming the equation to feature the variable alone.

Variable substitution is a fundamental technique in algebra and calculus used to simplify and solve equations or expressions. It involves replacing one variable with another expression or value, making the task of manipulation easier.
In our case, finding the inverse of \( f(x) = 3x + 5 \) uses variable substitution in two places:

• Initially, replacing \( f(x) \) with \( y \) to simplify the equation.
• Then swapping \( x \) and \( y \) in the equation \( y = 3x + 5 \) to start on isolating the original output variable as the new input variable.

Such substitutions help in removing clutter, making the equations more manageable to solve or rearrange. This technique is crucial for working with inverse functions where manipulating variables is necessary for expressing the function in terms of the input rather than the output.