91Ó°ÊÓ

In the realm of mathematics, substitution is a powerful technique utilized to simplify and solve equations. It involves replacing a variable with a given value or expression. In our exercise, understanding substitution is crucial. We are working with the function \( g(x) = -3x \). Our goal is to determine the outputs for \( g(w) \) and \( g(w+1) \). Here, \( x \) is the placeholder, and we carry out substitution by replacing \( x \) with \( w \) and \( w + 1 \) respectively.

• For \( g(w) \), substitute \( x = w \) into the equation, resulting in \( g(w) = -3w \).
• For \( g(w+1) \), substitute \( x = w + 1 \) into the equation, leading to \( g(w+1) = -3(w+1) = -3w - 3 \).

Substitution simplifies the process of finding function values and is frequently used in various mathematical contexts, especially in algebra. By switching out variables with specific values, we make equations more manageable and understandable.

In a function like \( g(x) = -3x \), the independent variable plays a pivotal role. It is the variable that we can change or control, and it directly influences the dependent variable. Here, \( x \) acts as the independent variable. In different scenarios, this could be any letter, serving as a stand-in for any real number.
Understanding the independent variable:

• It's where you input values. In our case, \( x \) could be replaced with numbers or expressions like \( w \) or \( w+1 \).
• The value of \( g(x) \) depends directly on the value assigned to \( x \).

By comprehending the function's configuration, we acknowledge that changes in the independent variable will directly modify the outcome of the function. This basic principle is essential for graphing functions and performing further calculations.

Function evaluation is the process of determining the output of a function based on specific input values of its independent variable. In our exercise, we evaluated the function \( g(x) = -3x \) for two different scenarios: \( g(w) \) and \( g(w+1) \).
Steps of function evaluation:

• Identify the function rule, which indicates how to calculate the result. Here, it's \(-3x\).
• Insert the chosen value or expression in place of the independent variable.
• Simplify or solve the expression to find the outcome.

For instance, substituting \( x = w \) gives \( g(w) = -3w \), while substituting \( x = w + 1 \) results in \( g(w+1) = -3w - 3 \). Function evaluation is commonly used to predict and examine function behavior across various inputs, providing valuable insights in both simple and complex mathematical circumstances.