Select The Correct Answer.Consider Functions $p$ And $q$.$\[ \begin{array}{l} p(x) = \log_2(x-1) \\ q(x) = 2^x - 1 \end{array} \\]Which Statement Is True About These Functions?A. The $x$-intercept Of Function

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Understanding the Properties of Functions p and q

In mathematics, functions are used to describe the relationship between variables. Two functions, p(x) and q(x), are given as:

p(x) = log2(x-1)

q(x) = 2^x - 1

To determine the correct statement about these functions, we need to analyze their properties and behavior.

Analyzing the x-Intercept of Function p(x)

The x-intercept of a function is the point where the function crosses the x-axis, i.e., where the value of the function is zero. To find the x-intercept of function p(x), we set p(x) = 0 and solve for x.

p(x) = log2(x-1) = 0

Using the properties of logarithms, we can rewrite the equation as:

2^0 = x - 1

Simplifying the equation, we get:

1 = x - 1

Adding 1 to both sides, we get:

x = 2

Therefore, the x-intercept of function p(x) is x = 2.

Analyzing the x-Intercept of Function q(x)

To find the x-intercept of function q(x), we set q(x) = 0 and solve for x.

q(x) = 2^x - 1 = 0

Adding 1 to both sides, we get:

2^x = 1

Since 2^x is always positive, the only solution to this equation is x = 0.

Therefore, the x-intercept of function q(x) is x = 0.

Comparing the x-Intercepts of Functions p and q

Now that we have found the x-intercepts of both functions, we can compare them.

The x-intercept of function p(x) is x = 2, while the x-intercept of function q(x) is x = 0.

Conclusion

Based on our analysis, we can conclude that the statement "The x-intercept of function p(x) is x = 2, while the x-intercept of function q(x) is x = 0" is true.

Answer

The correct answer is A.

Additional Analysis

To further analyze the properties of functions p and q, we can examine their domains and ranges.

Domain of Function p(x)

The domain of a function is the set of all possible input values for which the function is defined. For function p(x), the domain is all real numbers greater than 1, since the argument of the logarithm must be positive.

Range of Function p(x)

The range of a function is the set of all possible output values for which the function is defined. For function p(x), the range is all real numbers, since the logarithm can take on any real value.

Domain of Function q(x)

The domain of function q(x) is all real numbers, since the exponential function is defined for all real numbers.

Range of Function q(x)

The range of function q(x) is all real numbers greater than or equal to 0, since the exponential function is always positive.

Conclusion

In conclusion, functions p and q have different domains and ranges. Function p(x) is defined for all real numbers greater than 1, while function q(x) is defined for all real numbers. The range of function p(x) is all real numbers, while the range of function q(x) is all real numbers greater than or equal to 0.

Answer

The correct answer is A.

Final Conclusion

In this article, we analyzed the properties of functions p and q, including their x-intercepts, domains, and ranges. We concluded that the statement "The x-intercept of function p(x) is x = 2, while the x-intercept of function q(x) is x = 0" is true. We also examined the domains and ranges of both functions and found that they have different properties.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Functions, 2nd edition, James Stewart
  • [3] Mathematics, 2nd edition, Michael Artin

Note

This article is for educational purposes only and is not intended to be a comprehensive treatment of the subject.
Q&A: Functions p and q

In our previous article, we analyzed the properties of functions p and q, including their x-intercepts, domains, and ranges. In this article, we will answer some frequently asked questions about these functions.

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

A: The domain of function p(x) is all real numbers greater than 1, since the argument of the logarithm must be positive.

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

A: The range of function p(x) is all real numbers, since the logarithm can take on any real value.

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

A: The domain of function q(x) is all real numbers, since the exponential function is defined for all real numbers.

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

A: The range of function q(x) is all real numbers greater than or equal to 0, since the exponential function is always positive.

Q: How do the x-intercepts of functions p and q compare?

A: The x-intercept of function p(x) is x = 2, while the x-intercept of function q(x) is x = 0.

Q: What is the relationship between functions p and q?

A: Functions p and q are inverse functions, meaning that their composition is the identity function.

Q: How can we use functions p and q in real-world applications?

A: Functions p and q can be used in a variety of real-world applications, such as modeling population growth, chemical reactions, and electrical circuits.

Q: What are some common mistakes to avoid when working with functions p and q?

A: Some common mistakes to avoid when working with functions p and q include:

  • Not checking the domain and range of the functions
  • Not using the correct inverse function
  • Not considering the properties of the logarithm and exponential functions

Q: How can we graph functions p and q?

A: Functions p and q can be graphed using a variety of methods, including:

  • Using a graphing calculator
  • Plotting points on a coordinate plane
  • Using a graphing software

Q: What are some common applications of functions p and q in mathematics?

A: Functions p and q have a wide range of applications in mathematics, including:

  • Calculus: Functions p and q are used to model population growth, chemical reactions, and electrical circuits.
  • Algebra: Functions p and q are used to solve equations and inequalities.
  • Geometry: Functions p and q are used to model geometric shapes and transformations.

Q: How can we use functions p and q to model real-world phenomena?

A: Functions p and q can be used to model a variety of real-world phenomena, including:

  • Population growth: Functions p and q can be used to model the growth of a population over time.
  • Chemical reactions: Functions p and q can be used to model the rate of a chemical reaction.
  • Electrical circuits: Functions p and q can be used to model the behavior of electrical circuits.

Q: What are some common challenges when working with functions p and q?

A: Some common challenges when working with functions p and q include:

  • Understanding the properties of the logarithm and exponential functions
  • Using the correct inverse function
  • Considering the domain and range of the functions

Q: How can we overcome these challenges?

A: To overcome these challenges, it is essential to:

  • Understand the properties of the logarithm and exponential functions
  • Use the correct inverse function
  • Consider the domain and range of the functions

Conclusion

In this article, we answered some frequently asked questions about functions p and q. We hope that this article has provided you with a better understanding of these functions and their applications. If you have any further questions, please don't hesitate to ask.