Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{g}(\mathrm{g}(5)) $$

Step-by-Step Solution

Answer

The value of \( g(g(5)) \) is 1.

Step 1: Apply the function g to the inner value

We start by evaluating the function \(g\) at the given value of 5. The function \(g(x) = x - 2\) applies the operation of subtracting 2 from \(x\). So, calculate \(g(5)\) as follows: \[ g(5) = 5 - 2 = 3 \]

Step 2: Apply the function g to the result from Step 1

Next, take the result from Step 1 and apply the function \(g\) again. We found that \(g(5) = 3\). Now, apply \(g\) to this result:\[ g(3) = 3 - 2 = 1 \]

Key Concepts

Evaluating Functions

In mathematics, evaluating functions is like solving simple puzzles by substituting a given input value into a function to find the output. Each function has a specific operation defined for it. By knowing this operation, you can determine what the function's output will be for different input values.

Here's how it works:

• Understand the Function: Before you evaluate, you need to know what the function does. For example, given a function like \( g(x) = x - 2 \), it signifies that the function subtracts 2 from any input \( x \).
• Substitute the Input: Once you're clear about the function, substitute the input value into the place of \( x \). For \( g(5) \), you replace \( x \) with 5, making it \( 5 - 2 \).
• Calculate the Result: Perform the operation defined by the function. For \( g(5) = 5 - 2 \), the calculation would give you 3.

Understanding how to evaluate functions is crucial, especially when dealing with function compositions, where you have to evaluate a function several times, using the output of one evaluation as the input for the next.

Algebraic Functions

Algebraic functions are expressions constructed using algebraic operations such as addition, subtraction, multiplication, division, and exponentiation with constants and variables. They form a crucial part of algebra as they describe relationships between numbers and variables.

In the context of the original exercise, you are dealing with simple algebraic functions like \( f(x) = 3x \) and \( g(x) = x - 2 \). These two functions:

• Linear Functions: Both are examples of linear algebraic functions where the highest exponent of the variable is 1. The function \( f(x) = 3x \) scales its input by a factor of 3, while \( g(x) = x - 2 \) subtracts 2 from its input.
• Predictability: These functions are straightforward as they create a straight-line graph. This makes predicting the behavior or output of these functions relatively easy.

By learning to manipulate algebraic functions, you start to understand how variables interact through mathematical operations. This builds a foundation for solving more complex problems in mathematics.

Step-by-Step Solution

When tackling problems involving functions, using a step-by-step approach can be extremely beneficial. It helps break down complex operations into manageable pieces and ensures accuracy.

For the exercise given, the problem is to evaluate \( g(g(5)) \). Here's how the step-by-step method was utilized:

• Step 1: Evaluate \( g(5) \). Substitute 5 in the place of \( x \) in the function \( g(x) = x - 2 \). So, \( g(5) = 5 - 2 = 3 \).
• Step 2: Using the result from Step 1, evaluate \( g(3) \). Substitute 3 back into \( g(x) \), which results in \( g(3) = 3 - 2 = 1 \).

This method systematically takes you through the process of evaluating a composition of functions, making it easier to follow and understand. It’s a useful technique especially when handling not just one, but multiple layers of functions.