Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(y=x-2\)

Step-by-Step Solution

Answer

\( f(x) = x - 2 \); \( f(5) = 3 \).

Step 1: Identify the Form of the Function

The function given is in the form of a linear equation, specifically, it is given as \( y = x - 2 \). This equation can also be expressed as \( f(x) = x - 2 \) since \( y \) represents the value of the function at any point \( x \).

Step 2: Write the Expression for the Function

Given the function \( y = x - 2 \), writing it in the function notation, it becomes \( f(x) = x - 2 \).

Step 3: Substitute to Find Value at Specific Point

To find the value of the function at a specific point, substitute \( x = 5 \) into the function expression. So \( f(5) = 5 - 2 \).

Step 4: Simplify the Expression

Calculate the value by subtracting 2 from 5. This gives us \( f(5) = 3 \).

Key Concepts

Function Notation

Function notation is a crucial concept in mathematics, especially when dealing with linear equations. It provides a way to express a function's dependent variable in terms of its independent variable. Essentially, function notation replaces the typical "y" variable with "f(x)" to denote a function named 'f' depending on 'x'.

In the exercise, the linear equation is initially provided as \( y = x - 2 \). To convert this into function notation, replace 'y' with 'f(x)', leading to the equation \( f(x) = x - 2 \). This function notation indicates that for each input 'x', \( f(x) \) will deliver the corresponding value of the function.

Utilizing function notation is beneficial because it provides clarity by explicitly showing which variable is the input. It also makes it easier to differentiate between multiple functions, such as \( f(x) \) and \( g(x) \), when analyzing complex problems. This representation becomes increasingly important when dealing with multi-variable contexts or when functions are composed or compared.

Substitution

Substitution in mathematics refers to the process of replacing a variable with a given value or expression. In the context of functions, substitution allows us to find the specific output of a function for a particular input value.

In this exercise, substitution is used to find the output of the function \( f(x) = x - 2 \) when \( x = 5 \). The process involves inserting the value 5 wherever 'x' appears in the function:

• Start with the function \( f(x) = x - 2 \).
• Substitute the value 5 for 'x', resulting in \( f(5) = 5 - 2 \).

This method is straightforward and instrumental in determining specific values within a function, enabling us to explore how different inputs affect the outcome.

Evaluation of Functions

Evaluating a function involves calculating the actual value of the function expression after substituting a specific input value. This step follows substitution, aimed at simplifying the expression to reveal the function's output.

In this scenario, to find \( f(5) \), we have already substituted 5 into the function, resulting in the expression \( f(5) = 5 - 2 \). The next step is to simplify:

• Calculate the subtraction: \( 5 - 2 = 3 \).
• Thus, the value of the function for this input is \( f(5) = 3 \).

Evaluating functions is a fundamental concept, as it allows you to determine the precise output and understand how the function behaves under different conditions. It's an essential skill when solving and graphing equations or when performing mathematical modeling in various applications.