Problem 57 - Algebra 2

Question

Writing Evaluate \((g \circ f)(3),\) when \(f(x)=2 x\) and \(g(x)=x+1 .\) Explain what you do first and why.

Step-by-Step Solution

Answer

The evaluated result of the function composition \(g \circ f\)(3) where \(f(x)=2x\) and \(g(x)=x+1\) is 7.

Step 1: Evaluate the function \(f(3)\)

Plug the value of 3 into the function provided for \(f(x)\). The function for \(f(x)\) is \(2x\), we replace \(x\) with 3, resulting in \(f(3) = 2*3 = 6\).

Step 2: Evaluate the function \(g(f(3))\)

Next, we take the result of \(f(3)\), which is 6, and plug it into the function provided for \(g(x)\), which is \(x+1\). So we replace \(x\) with 6, resulting in \(g(f(3)) = g(6) = 6+1 = 7\).

Key Concepts

Evaluating Functions

Evaluating functions is all about figuring out what value you get when you plug a particular number into a function. Think of it as a way to answer the question: "If I put this number into the function, what comes out?" In the exercise given, you're first asked to evaluate the function \(f(x) = 2x\). This means you have to find \(f(3)\). To do this, you plug 3 into the function \(f(x)\), making it \(2 \times 3\).
This operation gives you a result of 6. This is the value that comes out when you evaluate the function \(f\) at 3.
The second function to evaluate is \(g(x) = x + 1\), and you use \(f(3)\) as the input, making it \(g(6)\). Taking the 6, you do \(6 + 1\), which equals 7. This final value is what you gain from evaluating the compound function, \((g \circ f)(3)\). Evaluating functions step by step, as shown here, is critical in math to ensure you're correctly working through problems.

Algebraic Operations

Algebraic operations are the steps you use to solve problems within equations or expressions. They include addition, subtraction, multiplication, and division. In our exercise, you are primarily using multiplication and addition.
To solve \(f(x) = 2x\), you multiply 2 by 3:

• Identify the operation needed: Multiplication in the case of \(2 \times 3\).
• Execute the operation: Multiply to get 6.

In \(g(x) = x + 1\), you perform an addition:

• Plug in the previous result 6 into \(g\).
• Add 1 to get 7.

By understanding these operations and how to perform them, such as replacing \(x\) and simplifying expressions into numbers, you can accurately evaluate and solve problems related to functions.

Step-by-Step Solutions

Step-by-step solutions guide you through the process of solving a problem one stage at a time. This method ensures a better understanding and minimizes the chances of making mistakes. You can think of it like a recipe, where each step must be followed in order to achieve the desired result.
In the original solution, the problem is broken down into manageable parts:

• First, you find \(f(3)\) by plugging 3 into \(f(x)\).
• Then, you calculate \(g(f(3))\) using the result from the first step.

This detailed breakdown makes it easier to follow the logic and understand how composite functions work. In math, this strategy is beneficial because it makes complex problems appear less daunting and builds a strong foundation for tackling more challenging material later on. Plus, it's a great way to get familiar with the order of operations and the importance of precision in calculations.