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Function evaluation is finding the output value of a function when a specific input is provided. This is an essential concept in mathematics as it allows us to understand how functions behave for different values of the variable. In our example, we need to find the value of the function \( f(x) = 2x + 3 \) when \( x = 5 \).

Here's how function evaluation works:

• Identify the given function. In this case, it's \( f(x) = 2x + 3 \).
• Determine the input value you need to evaluate. Our task is to find \( f(5) \).
• Substitute the input value \( x = 5 \) into the function, replacing all occurrences of \( x \) with \( 5 \).
• Perform the necessary arithmetic operations to find the result.
• Conclude with the output value which is the evaluation of the function at that input.

In the context of this example, substituting \( x = 5 \) in \( f(x) = 2x + 3 \) yields \( f(5) = 2(5) + 3 = 13 \). This value, \( 13 \), is the output of the function for the input \( 5 \).

Linear equations are mathematical expressions that represent straight lines when plotted on a two-dimensional graph. They are foundational in algebra and appear in many real-world contexts.

The general form of a linear equation is \( ax + b = 0 \), where \( a \) and \( b \) are constants. In our example, the function \( f(x) = 2x + 3 \) is a linear equation because it can be expressed in this standard form. Here:

• \( a = 2 \)
• \( b = 3 \)

Key characteristics of linear equations include:

• They have a constant rate of change, represented as the slope, which is \( 2 \) in this function.
• The graph of a linear equation is a straight line with a slope and a y-intercept. In \( f(x) = 2x + 3 \), the slope (rate of change) is \( 2 \), and the y-intercept (where the line crosses the y-axis) is \( 3 \).

Understanding linear equations is crucial because they serve as a simple yet powerful tool to model relationships and predict trends. The function's evaluation at specific points helps to determine its behavior along the line.

The substitution method is a technique used in mathematics to solve functions and equations. It involves replacing variables with numbers or other expressions to simplify and find solutions. This is particularly useful in evaluating functions and solving systems of equations.

In the function \( f(x) = 2x + 3 \), our goal was to find \( f(5) \).

• We use substitution by taking the value of \( x \), in this case, \( 5 \), and replacing \( x \) with \( 5 \) in the function.
• The function then becomes \( f(5) = 2(5) + 3 \).
• Performing the arithmetic operations gives us the value of \( f(5) \), which is \( 13 \).

This method is systematic and allows for clear and concise evaluation. When dealing with linear functions, the substitution method reveals the direct relationship between input variables and their corresponding outputs. This makes it easier to understand how changes in \( x \) affect the outcome of the function.