What Can You Say About The $y$-values Of The Two Functions $f(x)=3x^2-3$ And $g(x)=2^x-3$?A. $g(x)$ Has The Smallest Possible $y$-value.B. $f(x)$ And $g(x)$ Have Equivalent

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Introduction

When comparing two functions, it's essential to understand their behavior and how they relate to each other. In this case, we're given two functions, $f(x)=3x^2-3$ and $g(x)=2^x-3$, and we're asked to discuss their $y$-values. To approach this problem, we need to analyze the properties of each function and understand how they compare to each other.

Understanding the Functions

Let's start by understanding the properties of each function. The function $f(x)=3x^2-3$ is a quadratic function, which means it has a parabolic shape. This type of function has a single turning point, known as the vertex, and it opens upwards or downwards. In this case, the function opens upwards because the coefficient of the $x^2$ term is positive.

On the other hand, the function $g(x)=2^x-3$ is an exponential function. This type of function has a base, which is 2 in this case, and it grows rapidly as the input value increases. The function $g(x)$ has a horizontal asymptote at $y=-3$, which means that as $x$ approaches infinity, the function approaches $-3$.

Comparing the Functions

Now that we understand the properties of each function, let's compare them. We're asked to determine which function has the smallest possible $y$-value. To do this, we need to find the minimum value of each function.

For the function $f(x)=3x^2-3$, we can find the minimum value by completing the square or using calculus. By completing the square, we get:

f(x)=3(x2−1)f(x)=3(x^2-1)

This tells us that the minimum value of $f(x)$ occurs when $x^2-1=0$, which gives us $x=\pm1$. Plugging these values back into the function, we get:

f(−1)=3(−1)2−3=0f(-1)=3(-1)^2-3=0

f(1)=3(1)2−3=0f(1)=3(1)^2-3=0

So, the minimum value of $f(x)$ is $0$, which occurs when $x=\pm1$.

For the function $g(x)=2^x-3$, we can find the minimum value by analyzing its behavior. Since the function has a horizontal asymptote at $y=-3$, we know that the minimum value of $g(x)$ must be greater than or equal to $-3$. To find the exact minimum value, we can use calculus or numerical methods. By using calculus, we can find that the minimum value of $g(x)$ occurs when $x=-\infty$, which gives us a minimum value of $-3$.

Conclusion

In conclusion, we can say that the function $g(x)=2^x-3$ has the smallest possible $y$-value. This is because the minimum value of $g(x)$ is $-3$, which is less than the minimum value of $f(x)$. Therefore, option A is the correct answer.

Final Thoughts

When comparing two functions, it's essential to understand their properties and behavior. In this case, we analyzed the properties of the functions $f(x)=3x^2-3$ and $g(x)=2^x-3$ and compared their $y$-values. By understanding the properties of each function, we were able to determine which function has the smallest possible $y$-value. This type of analysis is essential in mathematics and has many real-world applications.

References

  • [1] Calculus: Early Transcendentals, James Stewart
  • [2] Algebra and Trigonometry, Michael Sullivan
  • [3] Functions, Graphs, and Analytic Geometry, Michael Spivak

Discussion

  • What are some other ways to compare two functions?
  • How do you determine the minimum value of a function?
  • What are some real-world applications of comparing functions?

Related Topics

  • Quadratic functions
  • Exponential functions
  • Calculus
  • Algebra
  • Trigonometry

Tags

  • Functions
  • Quadratic functions
  • Exponential functions
  • Calculus
  • Algebra
  • Trigonometry

Introduction

In our previous article, we discussed the properties of the functions $f(x)=3x^2-3$ and $g(x)=2^x-3$ and compared their $y$-values. We determined that the function $g(x)=2^x-3$ has the smallest possible $y$-value. In this article, we'll answer some frequently asked questions about the functions and their $y$-values.

Q&A

Q: What is the minimum value of the function $f(x)=3x^2-3$?

A: The minimum value of the function $f(x)=3x^2-3$ is $0$, which occurs when $x=\pm1$.

Q: What is the minimum value of the function $g(x)=2^x-3$?

A: The minimum value of the function $g(x)=2^x-3$ is $-3$, which occurs when $x=-\infty$.

Q: How do you compare the $y$-values of two functions?

A: To compare the $y$-values of two functions, you need to analyze their properties and behavior. You can use calculus or numerical methods to find the minimum value of each function.

Q: What are some real-world applications of comparing functions?

A: Comparing functions has many real-world applications, such as:

  • Modeling population growth
  • Analyzing economic data
  • Predicting weather patterns
  • Optimizing systems

Q: How do you determine the minimum value of a function?

A: To determine the minimum value of a function, you can use calculus or numerical methods. You can also use graphical methods, such as plotting the function and finding the minimum point.

Q: What are some other ways to compare two functions?

A: Some other ways to compare two functions include:

  • Using inequalities
  • Using optimization techniques
  • Using graphical methods
  • Using numerical methods

Conclusion

In conclusion, comparing functions is an essential skill in mathematics and has many real-world applications. By understanding the properties and behavior of functions, you can compare their $y$-values and determine which function has the smallest possible $y$-value. We hope this article has helped you understand the concepts and techniques involved in comparing functions.

Final Thoughts

Comparing functions is a complex topic that requires a deep understanding of mathematical concepts and techniques. However, with practice and experience, you can develop the skills and knowledge needed to compare functions and solve real-world problems.

References

  • [1] Calculus: Early Transcendentals, James Stewart
  • [2] Algebra and Trigonometry, Michael Sullivan
  • [3] Functions, Graphs, and Analytic Geometry, Michael Spivak

Discussion

  • What are some other ways to compare two functions?
  • How do you determine the minimum value of a function?
  • What are some real-world applications of comparing functions?

Related Topics

  • Quadratic functions
  • Exponential functions
  • Calculus
  • Algebra
  • Trigonometry

Tags

  • Functions
  • Quadratic functions
  • Exponential functions
  • Calculus
  • Algebra
  • Trigonometry

Additional Resources

  • Khan Academy: Functions and Graphs
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Functions and Graphs

FAQs

  • Q: What is the minimum value of the function $f(x)=3x^2-3$? A: The minimum value of the function $f(x)=3x^2-3$ is $0$, which occurs when $x=\pm1$.
  • Q: What is the minimum value of the function $g(x)=2^x-3$? A: The minimum value of the function $g(x)=2^x-3$ is $-3$, which occurs when $x=-\infty$.
  • Q: How do you compare the $y$-values of two functions? A: To compare the $y$-values of two functions, you need to analyze their properties and behavior. You can use calculus or numerical methods to find the minimum value of each function.