Consider These Functions:$\[ \begin{array}{l} f(x) = -2x - 1 \\ g(x) = -\frac{1}{2}x + \frac{1}{2} \end{array} \\]Which Statements, If Any, Are True About These Functions?I. The Function $f(g(x)) = X$ For All Real $x$.II. The

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Introduction

In mathematics, composite functions play a crucial role in various branches of study, including algebra, calculus, and analysis. A composite function is a function that is derived from two or more functions. In this article, we will explore the composite function $f(g(x))$, where $f(x) = -2x - 1$ and $g(x) = -\frac{1}{2}x + \frac{1}{2}$. We will examine the truth value of two statements regarding this composite function.

The Composite Function $f(g(x))$

To evaluate the composite function $f(g(x))$, we need to substitute the expression for $g(x)$ into the function $f(x)$. This means that we will replace $x$ in the function $f(x)$ with the expression $g(x)$.

f(g(x)) = -2(g(x)) - 1

Now, we can substitute the expression for $g(x)$ into the equation above:

f(g(x)) = -2\left(-\frac{1}{2}x + \frac{1}{2}\right) - 1

Simplifying the expression, we get:

f(g(x)) = x - 1

Evaluating Statement I

Statement I claims that the function $f(g(x)) = x$ for all real $x$. However, based on our evaluation of the composite function $f(g(x))$, we can see that:

f(g(x)) = x - 1

This means that statement I is false, as the composite function $f(g(x))$ is not equal to $x$ for all real $x$.

Evaluating Statement II

Statement II claims that the function $f(g(x))$ is a one-to-one function. A one-to-one function is a function that maps distinct elements of its domain to distinct elements of its range. To determine if the function $f(g(x))$ is one-to-one, we need to examine its derivative.

The derivative of the function $f(g(x))$ is:

f'(g(x)) = 1

Since the derivative of the function $f(g(x))$ is a constant, the function is not one-to-one. Therefore, statement II is also false.

Conclusion

In conclusion, we have evaluated the composite function $f(g(x))$, where $f(x) = -2x - 1$ and $g(x) = -\frac{1}{2}x + \frac{1}{2}$. We have found that statement I is false, as the composite function $f(g(x))$ is not equal to $x$ for all real $x$. We have also found that statement II is false, as the function $f(g(x))$ is not one-to-one.

Key Takeaways

• The composite function $f(g(x))$ is not equal to $x$ for all real $x$.
• The function $f(g(x))$ is not one-to-one.
• The derivative of the function $f(g(x))$ is a constant.

Further Reading

For more information on composite functions and one-to-one functions, we recommend the following resources:

• Khan Academy: Composite Functions
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: One-to-One Functions

References

• [1] Thomas, G. B., & Finney, R. E. (2014). Calculus and Analytic Geometry. Pearson Education.
• [2] Larson, R. E., & Edwards, B. I. (2013). Calculus: Early Transcendentals. Cengage Learning.
• [3] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  Frequently Asked Questions: Composite Functions and One-to-One Functions ====================================================================

Q: What is a composite function?

A: A composite function is a function that is derived from two or more functions. It is formed by substituting one function into another function.

Q: How do you evaluate a composite function?

A: To evaluate a composite function, you need to substitute the expression for the inner function into the outer function. This means that you will replace the variable in the outer function with the expression for the inner function.

Q: What is a one-to-one function?

A: A one-to-one function is a function that maps distinct elements of its domain to distinct elements of its range. In other words, a one-to-one function is a function that passes the horizontal line test.

Q: How do you determine if a function is one-to-one?

A: To determine if a function is one-to-one, you need to examine its derivative. If the derivative is a constant, then the function is not one-to-one.

Q: What is the significance of one-to-one functions?

A: One-to-one functions are significant because they have an inverse function. This means that if you have a one-to-one function, you can find its inverse function by swapping the x and y variables.

Q: Can you give an example of a composite function?

A: Yes, here is an example of a composite function:

Let $f(x) = 2x + 1$ and $g(x) = x^2$. Then, the composite function $f(g(x))$ is:

$f(g(x)) = 2(g(x)) + 1$ $f(g(x)) = 2(x^2) + 1$ $f(g(x)) = 2x^2 + 1$

Q: Can you give an example of a one-to-one function?

A: Yes, here is an example of a one-to-one function:

Let $f(x) = 2x + 1$. Then, the function $f(x)$ is one-to-one because it passes the horizontal line test.

Q: What is the relationship between composite functions and one-to-one functions?

A: Composite functions and one-to-one functions are related because if a composite function is one-to-one, then it has an inverse function. This means that if you have a composite function that is one-to-one, you can find its inverse function by swapping the x and y variables.

Q: Can you give an example of a composite function that is not one-to-one?

A: Yes, here is an example of a composite function that is not one-to-one:

Let $f(x) = 2x + 1$ and $g(x) = x^2$. Then, the composite function $f(g(x))$ is:

$f(g(x)) = 2(g(x)) + 1$ $f(g(x)) = 2(x^2) + 1$ $f(g(x)) = 2x^2 + 1$

This composite function is not one-to-one because its derivative is a constant.

Q: What is the significance of composite functions and one-to-one functions in real-world applications?

A: Composite functions and one-to-one functions are significant in real-world applications because they are used to model complex systems and relationships. For example, in economics, composite functions are used to model the relationship between supply and demand. In physics, one-to-one functions are used to model the relationship between energy and momentum.

Conclusion

In conclusion, composite functions and one-to-one functions are important concepts in mathematics that have significant applications in real-world problems. By understanding these concepts, you can better model complex systems and relationships, and make more informed decisions in various fields.

Key Takeaways

• Composite functions are formed by substituting one function into another function.
• One-to-one functions are functions that map distinct elements of their domain to distinct elements of their range.
• Composite functions and one-to-one functions are related because if a composite function is one-to-one, then it has an inverse function.
• Composite functions and one-to-one functions have significant applications in real-world problems.

Further Reading

For more information on composite functions and one-to-one functions, we recommend the following resources:

• Khan Academy: Composite Functions
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: One-to-One Functions

References

• [1] Thomas, G. B., & Finney, R. E. (2014). Calculus and Analytic Geometry. Pearson Education.
• [2] Larson, R. E., & Edwards, B. I. (2013). Calculus: Early Transcendentals. Cengage Learning.
• [3] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.