Select The Correct Answer.If F ( X ) = 4 X F(x) = 4^x F ( X ) = 4 X , Then G ( X ) = ? G(x) = ? G ( X ) = ? Which Of The Following Is Equal To G ( X G(x G ( X ]?A. 4 ( X − 2 ) 4^{(x-2)} 4 ( X − 2 ) B. 4 X + 2 4^x + 2 4 X + 2 C. 4 X − 2 4^x - 2 4 X − 2 D. 4 ( X + 2 ) 4^{(x+2)} 4 ( X + 2 )

Introduction

In mathematics, functions are used to describe relationships between variables. Given a function, we can manipulate it to create new functions. In this article, we will explore how to find the value of g(x) given the function f(x) = 4^x.

Understanding the Function f(x)

The function f(x) = 4^x is an exponential function. This means that as x increases, the value of f(x) increases exponentially. For example, if x = 2, then f(x) = 4^2 = 16. If x = 3, then f(x) = 4^3 = 64.

Manipulating the Function f(x)

To find g(x), we need to manipulate the function f(x) = 4^x. We can do this by using algebraic operations such as addition, subtraction, multiplication, and division.

Option A: $4^{(x-2)}$

Let's start by analyzing option A: $4^{(x-2)}$. To determine if this is equal to g(x), we need to manipulate the function f(x) = 4^x to match this expression.

We can rewrite f(x) = 4^x as f(x) = 4^(x-2+2). Using the properties of exponents, we can simplify this expression to f(x) = 4^(x-2) * 4^2.

Since 4^2 = 16, we can rewrite f(x) = 4^(x-2) * 16. This is equivalent to f(x) = 16 * 4^(x-2).

Now, let's compare this expression to option A: $4^{(x-2)}$. We can see that they are equivalent, but with a constant factor of 16.

Option B: $4^x + 2$

Let's analyze option B: $4^x + 2$. To determine if this is equal to g(x), we need to manipulate the function f(x) = 4^x to match this expression.

We can rewrite f(x) = 4^x as f(x) = 4^x + 0. However, this is not equivalent to option B, since the constant term is 0, not 2.

Option C: $4^x - 2$

Let's analyze option C: $4^x - 2$. To determine if this is equal to g(x), we need to manipulate the function f(x) = 4^x to match this expression.

We can rewrite f(x) = 4^x as f(x) = 4^x - 0. However, this is not equivalent to option C, since the constant term is 0, not -2.

Option D: $4^{(x+2)}$

Let's analyze option D: $4^{(x+2)}$. To determine if this is equal to g(x), we need to manipulate the function f(x) = 4^x to match this expression.

We can rewrite f(x) = 4^x as f(x) = 4^(x+2-2). Using the properties of exponents, we can simplify this expression to f(x) = 4^(x+2) / 4^2.

Since 4^2 = 16, we can rewrite f(x) = 4^(x+2) / 16. This is not equivalent to option D, since the expression is divided by 16, not multiplied by 16.

Conclusion

Based on our analysis, we can see that option A: $4^{(x-2)}$ is the correct answer. This is because we can manipulate the function f(x) = 4^x to match this expression, with a constant factor of 16.

Therefore, the correct answer is:

A. $4^{(x-2)}$

Final Answer

Introduction

In our previous article, we explored how to find the value of g(x) given the function f(x) = 4^x. We analyzed four different options and determined that option A: $4^{(x-2)}$ is the correct answer.

In this article, we will provide a Q&A section to help clarify any questions or concerns you may have about the function g(x).

Q: What is the relationship between f(x) and g(x)?

A: The function g(x) is a transformation of the function f(x) = 4^x. Specifically, g(x) is equivalent to f(x-2).

Q: How do I manipulate the function f(x) to match g(x)?

A: To manipulate the function f(x) to match g(x), you can use algebraic operations such as addition, subtraction, multiplication, and division. In this case, we can rewrite f(x) = 4^x as f(x) = 4^(x-2+2) and simplify it to f(x) = 4^(x-2) * 4^2.

Q: What is the significance of the constant factor 16 in option A?

A: The constant factor 16 is not part of the function g(x). It is simply a result of the manipulation of the function f(x) to match g(x). In other words, g(x) = 4^(x-2) is equivalent to f(x) = 16 * 4^(x-2).

Q: Can I use other algebraic operations to manipulate the function f(x)?

A: Yes, you can use other algebraic operations to manipulate the function f(x). However, you must ensure that the resulting expression is equivalent to g(x).

Q: How do I determine if an expression is equivalent to g(x)?

A: To determine if an expression is equivalent to g(x), you can use the following steps:

1. Rewrite the expression in terms of the function f(x) = 4^x.
2. Simplify the expression using algebraic operations.
3. Compare the resulting expression to g(x).

Q: Can I use other functions to find g(x)?

A: Yes, you can use other functions to find g(x). However, you must ensure that the resulting expression is equivalent to g(x).

Q: What are some common mistakes to avoid when finding g(x)?

A: Some common mistakes to avoid when finding g(x) include:

• Not rewriting the expression in terms of the function f(x) = 4^x.
• Not simplifying the expression using algebraic operations.
• Not comparing the resulting expression to g(x).

Conclusion

In this Q&A article, we provided answers to common questions about the function g(x). We hope that this article has helped clarify any questions or concerns you may have had about the function g(x).

Final Answer

The final answer is A. $4^{(x-2)}$.