91Ó°ÊÓ

A linear function is a foundational concept in algebra and precalculus that represents a straight line when graphed. It is defined by an equation of the form
\( y = mx + b \),
where m is the slope of the line, and b is the y-intercept. The slope indicates how steep the line is, and the y-intercept specifies where the line crosses the y-axis.

For the function
\( f(x) = \frac{2}{3}x - 1 \),
\( m = \frac{2}{3} \) is the slope, showing that for each unit increase in x, the function value increases by \( \frac{2}{3} \) units. The term <\( -1 \) is the y-intercept, telling us where the function crosses the y-axis.

Understanding these components of the linear function allows students to predict the behavior of the function and calculate its values for given inputs.

The process of substituting values is crucial when evaluating functions. It involves replacing the variable in the function's expression with a given number. The main goal is to find the output of the function, known as the function value, for that particular input.

To correctly substitute a value, follow these steps:

• Identify the variable in the function's formula.
• Replace the variable with the given value.
• Ensure that the substitution is clear and follows the order of operations.

For example, substituting x with 3 in the function \( f(x) = \frac{2}{3}x - 1 \) involves rewriting the expression as \( f(3) = \frac{2}{3} \cdot 3 - 1 \) and then simplifying it. By diligently performing substitutions, students can evaluate the function at any point, facilitating a deeper understanding of the function's behavior.

Once values have been substituted into a function, the next step is simplifying expressions. Simplification is the process of reducing an expression to its most basic form, making it easier to understand or solve. In the context of functions, simplification includes arithmetic operations such as adding, subtracting, multiplying, and dividing. It's also important to simplify fractional or complex expressions.

Following the correct order of operations is crucial for proper simplification. Remember the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)—to ensure you simplify in the right order.

For instance, with the function \( f(3) = \frac{2}{3} \cdot 3 - 1 \), multiplication comes before subtraction, resulting in \( f(3) = 2 - 1 = 1 \). Simplifying expressions accurately is vital to obtaining the correct function values and mastering the evaluation of functions.