Simplify The Function:$\[ G(1-t) = 2(1-t)^2 - 3(1-t) + 8 \\]

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Introduction

In mathematics, simplifying functions is an essential skill that helps us understand and work with complex equations. In this article, we will focus on simplifying the given function: g(1−t)=2(1−t)2−3(1−t)+8g(1-t) = 2(1-t)^2 - 3(1-t) + 8. We will break down the process into manageable steps, making it easier to understand and follow along.

Understanding the Function

Before we start simplifying the function, let's take a closer look at what it represents. The function g(1−t)g(1-t) is a quadratic function, which means it has a squared term. The function is defined as g(1−t)=2(1−t)2−3(1−t)+8g(1-t) = 2(1-t)^2 - 3(1-t) + 8. Our goal is to simplify this function, making it easier to work with.

Step 1: Expand the Squared Term

To simplify the function, we need to start by expanding the squared term. The squared term is (1−t)2(1-t)^2. To expand this term, we need to multiply the binomial (1−t)(1-t) by itself.

(1−t)2=(1−t)(1−t)(1-t)^2 = (1-t)(1-t)

Using the FOIL method (First, Outer, Inner, Last), we can multiply the two binomials:

(1−t)(1−t)=12−2(1)(t)+t2(1-t)(1-t) = 1^2 - 2(1)(t) + t^2

Simplifying the expression, we get:

(1−t)2=1−2t+t2(1-t)^2 = 1 - 2t + t^2

Step 2: Substitute the Expanded Squared Term

Now that we have expanded the squared term, we can substitute it back into the original function. The original function is g(1−t)=2(1−t)2−3(1−t)+8g(1-t) = 2(1-t)^2 - 3(1-t) + 8. We will replace the squared term with the expanded expression we found in Step 1.

g(1−t)=2(1−2t+t2)−3(1−t)+8g(1-t) = 2(1 - 2t + t^2) - 3(1-t) + 8

Step 3: Distribute the Coefficients

Next, we need to distribute the coefficients to the terms inside the parentheses. The coefficient 2 is multiplied by the terms inside the parentheses:

2(1−2t+t2)=2−4t+2t22(1 - 2t + t^2) = 2 - 4t + 2t^2

The coefficient -3 is multiplied by the term inside the parentheses:

−3(1−t)=−3+3t-3(1-t) = -3 + 3t

Step 4: Combine Like Terms

Now that we have distributed the coefficients, we can combine like terms. The like terms are the terms that have the same variable and exponent. In this case, the like terms are the terms with the variable t.

g(1−t)=2−4t+2t2−3+3t+8g(1-t) = 2 - 4t + 2t^2 - 3 + 3t + 8

Combining the like terms, we get:

g(1−t)=2−3+8−4t+3t+2t2g(1-t) = 2 - 3 + 8 - 4t + 3t + 2t^2

Simplifying the expression, we get:

g(1−t)=7−t+2t2g(1-t) = 7 - t + 2t^2

Conclusion

In this article, we simplified the function g(1−t)=2(1−t)2−3(1−t)+8g(1-t) = 2(1-t)^2 - 3(1-t) + 8. We broke down the process into manageable steps, making it easier to understand and follow along. We expanded the squared term, substituted it back into the original function, distributed the coefficients, and combined like terms. The simplified function is g(1−t)=7−t+2t2g(1-t) = 7 - t + 2t^2. This function is now easier to work with, and we can use it to solve problems and make predictions.

Final Answer

Introduction

In our previous article, we simplified the function g(1−t)=2(1−t)2−3(1−t)+8g(1-t) = 2(1-t)^2 - 3(1-t) + 8. We broke down the process into manageable steps, making it easier to understand and follow along. In this article, we will answer some common questions related to simplifying functions.

Q: What is the purpose of simplifying functions?

A: The purpose of simplifying functions is to make them easier to work with. Simplifying functions helps us to understand and analyze complex equations, making it easier to solve problems and make predictions.

Q: What are some common techniques used to simplify functions?

A: Some common techniques used to simplify functions include:

  • Expanding squared terms
  • Distributing coefficients
  • Combining like terms
  • Canceling out common factors

Q: How do I know when to simplify a function?

A: You should simplify a function when:

  • The function is complex and difficult to work with
  • You need to analyze or solve the function
  • You need to make predictions or draw conclusions based on the function

Q: What are some common mistakes to avoid when simplifying functions?

A: Some common mistakes to avoid when simplifying functions include:

  • Forgetting to distribute coefficients
  • Not combining like terms
  • Canceling out common factors incorrectly
  • Not checking for errors in the simplified function

Q: Can I simplify functions with variables?

A: Yes, you can simplify functions with variables. In fact, simplifying functions with variables is often more challenging than simplifying functions with constants. However, the techniques used to simplify functions with variables are similar to those used to simplify functions with constants.

Q: How do I know if a function is already simplified?

A: You can check if a function is already simplified by:

  • Checking if the function is in its simplest form
  • Checking if the function has any common factors that can be canceled out
  • Checking if the function has any like terms that can be combined

Q: Can I use technology to simplify functions?

A: Yes, you can use technology to simplify functions. Many graphing calculators and computer algebra systems (CAS) can simplify functions automatically. However, it's still important to understand the techniques used to simplify functions, as technology can sometimes make mistakes.

Conclusion

In this article, we answered some common questions related to simplifying functions. We discussed the purpose of simplifying functions, common techniques used to simplify functions, and common mistakes to avoid. We also discussed how to simplify functions with variables and how to check if a function is already simplified. By understanding these concepts, you can become more confident and proficient in simplifying functions.

Final Answer

The final answer is that simplifying functions is an essential skill in mathematics that helps us understand and work with complex equations. By understanding the techniques used to simplify functions, you can become more confident and proficient in simplifying functions and solving problems.

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