Function \[$ G \$\] Is Represented By The Equation:$\[ G(x) = 4\left(\frac{1}{4}\right)^x + 2 \\]Which Statement Correctly Compares The Two Functions?A. They Have The Same \[$ Y \$\]-intercept And The Same End Behavior. B.

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Introduction

In mathematics, functions are used to describe the relationship between two variables. When comparing two functions, it's essential to analyze their properties, such as the y-intercept, end behavior, and other characteristics. In this article, we will compare two functions, represented by the equations g(x)=4(14)x+2g(x) = 4\left(\frac{1}{4}\right)^x + 2 and f(x)=2(12)x+1f(x) = 2\left(\frac{1}{2}\right)^x + 1. We will examine their y-intercepts, end behaviors, and other properties to determine which statement correctly compares the two functions.

The Functions

Let's start by analyzing the given functions:

  • g(x)=4(14)x+2g(x) = 4\left(\frac{1}{4}\right)^x + 2
  • f(x)=2(12)x+1f(x) = 2\left(\frac{1}{2}\right)^x + 1

Y-Intercept

The y-intercept of a function is the point where the function intersects the y-axis. To find the y-intercept, we set x = 0 and solve for y.

For function g(x):

g(0)=4(14)0+2g(0) = 4\left(\frac{1}{4}\right)^0 + 2 g(0)=4(1)+2g(0) = 4(1) + 2 g(0)=6g(0) = 6

For function f(x):

f(0)=2(12)0+1f(0) = 2\left(\frac{1}{2}\right)^0 + 1 f(0)=2(1)+1f(0) = 2(1) + 1 f(0)=3f(0) = 3

End Behavior

The end behavior of a function describes how the function behaves as x approaches positive or negative infinity. To analyze the end behavior, we can examine the leading term of the function.

For function g(x):

g(x)=4(14)x+2g(x) = 4\left(\frac{1}{4}\right)^x + 2 The leading term is 4(14)x4\left(\frac{1}{4}\right)^x, which approaches 0 as x approaches positive or negative infinity.

For function f(x):

f(x)=2(12)x+1f(x) = 2\left(\frac{1}{2}\right)^x + 1 The leading term is 2(12)x2\left(\frac{1}{2}\right)^x, which approaches 0 as x approaches positive or negative infinity.

Comparing the Functions

Now that we have analyzed the y-intercepts and end behaviors of the two functions, let's compare them.

  • Y-Intercept: The y-intercept of function g(x) is 6, while the y-intercept of function f(x) is 3. Therefore, the two functions do not have the same y-intercept.
  • End Behavior: Both functions have the same end behavior, approaching 0 as x approaches positive or negative infinity.

Conclusion

Based on our analysis, we can conclude that the two functions do not have the same y-intercept. However, they do have the same end behavior. Therefore, the correct statement is:

  • A. They have the same end behavior.

However, the statement also mentions that they have the same y-intercept, which is incorrect. Therefore, the correct statement is not among the options provided.

Additional Analysis

Let's analyze the functions further to see if we can find any other properties that are the same.

  • Domain: The domain of a function is the set of all possible input values. For function g(x), the domain is all real numbers, while for function f(x), the domain is also all real numbers.
  • Range: The range of a function is the set of all possible output values. For function g(x), the range is all real numbers greater than or equal to 2, while for function f(x), the range is all real numbers greater than or equal to 1.

Conclusion

Based on our analysis, we can conclude that the two functions have the same end behavior, but they do not have the same y-intercept. They also have the same domain and range. Therefore, the correct statement is:

  • A. They have the same end behavior.

However, the statement also mentions that they have the same y-intercept, which is incorrect. Therefore, the correct statement is not among the options provided.

Final Conclusion

In conclusion, the two functions have the same end behavior, but they do not have the same y-intercept. They also have the same domain and range. Therefore, the correct statement is:

  • A. They have the same end behavior.

However, the statement also mentions that they have the same y-intercept, which is incorrect. Therefore, the correct statement is not among the options provided.

References

Additional Resources

FAQs

  • Q: What is the y-intercept of function g(x)? A: The y-intercept of function g(x) is 6.
  • Q: What is the end behavior of function g(x)? A: The end behavior of function g(x) is that it approaches 0 as x approaches positive or negative infinity.
  • Q: What is the domain of function g(x)? A: The domain of function g(x) is all real numbers.
  • Q: What is the range of function g(x)? A: The range of function g(x) is all real numbers greater than or equal to 2.
    Function Comparison Q&A ==========================

Introduction

In our previous article, we compared two functions, represented by the equations g(x)=4(14)x+2g(x) = 4\left(\frac{1}{4}\right)^x + 2 and f(x)=2(12)x+1f(x) = 2\left(\frac{1}{2}\right)^x + 1. We analyzed their y-intercepts, end behaviors, and other properties to determine which statement correctly compares the two functions. In this article, we will provide a Q&A section to answer some of the most frequently asked questions about the comparison of the two functions.

Q&A

Q: What is the y-intercept of function g(x)?

A: The y-intercept of function g(x) is 6.

Q: What is the end behavior of function g(x)?

A: The end behavior of function g(x) is that it approaches 0 as x approaches positive or negative infinity.

Q: What is the domain of function g(x)?

A: The domain of function g(x) is all real numbers.

Q: What is the range of function g(x)?

A: The range of function g(x) is all real numbers greater than or equal to 2.

Q: Do the two functions have the same y-intercept?

A: No, the two functions do not have the same y-intercept. The y-intercept of function g(x) is 6, while the y-intercept of function f(x) is 3.

Q: Do the two functions have the same end behavior?

A: Yes, the two functions have the same end behavior. Both functions approach 0 as x approaches positive or negative infinity.

Q: Do the two functions have the same domain?

A: Yes, the two functions have the same domain. Both functions have a domain of all real numbers.

Q: Do the two functions have the same range?

A: No, the two functions do not have the same range. The range of function g(x) is all real numbers greater than or equal to 2, while the range of function f(x) is all real numbers greater than or equal to 1.

Q: What is the difference between the two functions?

A: The main difference between the two functions is their y-intercept. The y-intercept of function g(x) is 6, while the y-intercept of function f(x) is 3. Additionally, the range of function g(x) is all real numbers greater than or equal to 2, while the range of function f(x) is all real numbers greater than or equal to 1.

Q: Can you provide an example of how to compare two functions?

A: Yes, here is an example of how to compare two functions:

Suppose we have two functions, f(x) = 2x + 1 and g(x) = 3x - 2. To compare these functions, we can analyze their y-intercepts, end behaviors, and other properties.

  • Y-Intercept: The y-intercept of function f(x) is 1, while the y-intercept of function g(x) is -2.
  • End Behavior: The end behavior of function f(x) is that it approaches positive infinity as x approaches positive infinity, while the end behavior of function g(x) is that it approaches negative infinity as x approaches positive infinity.
  • Domain: The domain of both functions is all real numbers.
  • Range: The range of function f(x) is all real numbers greater than or equal to 1, while the range of function g(x) is all real numbers less than or equal to -2.

Based on this analysis, we can conclude that the two functions have different y-intercepts, end behaviors, and ranges.

Q: Can you provide a real-world example of how to compare two functions?

A: Yes, here is a real-world example of how to compare two functions:

Suppose we have two functions, f(x) = 2x + 1 and g(x) = 3x - 2, that represent the cost of producing x units of a product. To compare these functions, we can analyze their y-intercepts, end behaviors, and other properties.

  • Y-Intercept: The y-intercept of function f(x) represents the fixed cost of producing the product, which is $1. The y-intercept of function g(x) represents the fixed cost of producing the product, which is $-2.
  • End Behavior: The end behavior of function f(x) represents the cost of producing x units of the product as x approaches positive infinity, which is $2x + 1. The end behavior of function g(x) represents the cost of producing x units of the product as x approaches positive infinity, which is $3x - 2.
  • Domain: The domain of both functions is all real numbers, representing the number of units of the product that can be produced.
  • Range: The range of function f(x) represents the cost of producing x units of the product, which is all real numbers greater than or equal to $1. The range of function g(x) represents the cost of producing x units of the product, which is all real numbers less than or equal to $-2.

Based on this analysis, we can conclude that the two functions have different y-intercepts, end behaviors, and ranges, which can help us make informed decisions about the production of the product.

Conclusion

In conclusion, the comparison of two functions involves analyzing their y-intercepts, end behaviors, and other properties. By understanding these properties, we can make informed decisions about the functions and their applications. We hope this Q&A article has provided you with a better understanding of how to compare two functions.