Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ g \circ f(-2) $$

Step-by-Step Solution

Answer

The composition \( g \circ f(-2) \) evaluates to \(-8\).

Step 1: Understand the Composition

The composition function \( g \circ f(x) \) indicates that we need to first apply function \( f \) and then apply function \( g \) to the result of \( f \). This is written as \( g(f(x)) \).

Step 2: Apply Function f

Substitute \( x = -2 \) into function \( f(x) = 3x \) to find \( f(-2) \). This gives us: \[f(-2) = 3(-2) = -6\]

Step 3: Apply Function g

Use the result from Step 2 and substitute \( f(-2) = -6 \) into function \( g(x) = x - 2 \) to find \( g(-6) \). This results in: \[g(-6) = -6 - 2 = -8\]

Step 4: Conclusion

The value of the composition \( g \circ f(-2) \) is \(-8\).

Key Concepts

Composition of Functions

Function composition can be thought of as the process of plugging one function into another. Consider the functions \( f(x) \) and \( g(x) \). To find the composition \( g \circ f(x) \), you will first apply function \( f \) to your input \( x \), and then use the result as the input to function \( g \).
Here is how you approach a composition of functions:

• Identify the inner and outer functions. In this case, \( f(x) \) is the inner function and \( g(x) \) is the outer function.
• Compute the result of the inner function using the given input. For example, if asked for \( g \circ f(-2) \), you calculate \( f(-2) \) first.
• Use the output from the inner function as the input for the outer function.

The result is the final output of the composition formula. Understand that the order matters in composition, so \( g \circ f(x) \) might not equal \( f \circ g(x) \).

Evaluating Functions

Evaluating a function involves substituting a specific value into the function and solving the equation that results. When working with function compositions, evaluating means applying this practice twice—once for the inner function and once for the outer.
Let’s break it down step by step:

• Evaluate the Inner Function: Substitute the given input into the inner function. For \( f(x) = 3x \), if \( x = -2 \), then \( f(-2) = 3(-2) = -6 \). You find the output for \( f \) first.
• Evaluate the Outer Function: Use the result from the inner function as the input for the outer function. With \( g(x) = x - 2 \), you substitute the result \( -6 \), giving you \( g(-6) = -6 - 2 = -8 \).

The outcome reflects how input values travel through each function sequentially.

Algebra 2 Concepts

Within Algebra 2, function composition is a valuable skill that combines several algebraic concepts.
Understanding function composition is critical for developing these key skills:

• Function Notation: Knowing how to work with functions using notation like \( f(x) \) ensures your calculations remain organized and logical.
• Order of Operations: Function composition solidifies the importance of performing operations in a specific sequence. This sequencing is comparable to applying parentheses in mathematical expressions.
• Input and Output Relationships: Algebra 2 emphasizes understanding how inputs (domains) and outputs (ranges) interact in functions, which applies directly to function composition.

Overall, function compositions require careful analysis and bolster a deep comprehension of foundational algebraic principles.