Select The Correct Answer.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 F(g(x)) = X F ( G ( X )) = X For All

Understanding Function Composition

In mathematics, function composition is a process of combining two or more functions to create a new function. This new function takes the output of one function as the input for another function. In this article, we will explore the composition of two given functions, $f(x)$ and $g(x)$, and determine which statements are true about these functions.

Defining the Functions

The two functions given are:

{ \begin{array}{l} f(x) = -2x - 1 \\ g(x) = -\frac{1}{2}x + \frac{1}{2} \end{array} \}

Evaluating the Composition of Functions

To evaluate the composition of functions, we need to substitute the output of one function as the input for another function. In this case, we need to find $f(g(x))$.

Calculating $f(g(x))$

To calculate $f(g(x))$, we need to substitute $g(x)$ into $f(x)$.

{ \begin{align*} f(g(x)) &= f\left(-\frac{1}{2}x + \frac{1}{2}\right) \\ &= -2\left(-\frac{1}{2}x + \frac{1}{2}\right) - 1 \\ &= x - 1 - 1 \\ &= x - 2 \end{align*} \}

Analyzing the Results

From the calculation above, we can see that $f(g(x)) = x - 2$. This means that the composition of functions $f(g(x))$ is not equal to $x$ for all values of $x$. Therefore, statement I is false.

Evaluating Statement II

Statement II claims that $g(f(x)) = x$ for all values of $x$. To evaluate this statement, we need to calculate $g(f(x))$.

Calculating $g(f(x))$

To calculate $g(f(x))$, we need to substitute $f(x)$ into $g(x)$.

{ \begin{align*} g(f(x)) &= g(-2x - 1) \\ &= -\frac{1}{2}(-2x - 1) + \frac{1}{2} \\ &= x + \frac{1}{2} - \frac{1}{2} \\ &= x \end{align*} \}

Analyzing the Results

From the calculation above, we can see that $g(f(x)) = x$. This means that statement II is true.

Conclusion

In conclusion, we have evaluated the composition of two functions, $f(x)$ and $g(x)$, and determined which statements are true about these functions. We found that statement I is false, while statement II is true. This analysis demonstrates the importance of understanding function composition in mathematics.

Key Takeaways

• Function composition is a process of combining two or more functions to create a new function.
• The composition of functions can be evaluated by substituting the output of one function as the input for another function.
• In this case, we found that $f(g(x)) = x - 2$, while $g(f(x)) = x$.

Final Thoughts

Function composition is a fundamental concept in mathematics, and understanding it is crucial for solving problems in various fields, including algebra, calculus, and engineering. By analyzing the composition of functions, we can gain insights into the behavior of complex systems and make informed decisions.

Understanding Function Composition

In our previous article, we explored the composition of two functions, $f(x)$ and $g(x)$, and determined which statements are true about these functions. In this article, we will answer some frequently asked questions about function composition.

Q: What is function composition?

A: Function composition is a process of combining two or more functions to create a new function. This new function takes the output of one function as the input for another function.

Q: Why is function composition important?

A: Function composition is important because it allows us to create new functions from existing functions. This can be useful in a variety of applications, including algebra, calculus, and engineering.

Q: How do I evaluate the composition of functions?

A: To evaluate the composition of functions, you need to substitute the output of one function as the input for another function. This can be done using the following steps:

1. Substitute the output of one function into the other function.
2. Simplify the resulting expression.
3. Evaluate the final expression.

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

A: $f(g(x))$ and $g(f(x))$ are two different compositions of functions. $f(g(x))$ means that we first apply the function $g(x)$ and then apply the function $f(x)$ to the result. $g(f(x))$ means that we first apply the function $f(x)$ and then apply the function $g(x)$ to the result.

Q: Can I always find a function that is the inverse of another function?

A: No, not always. In some cases, it may not be possible to find a function that is the inverse of another function. This is because the inverse function may not exist or may not be unique.

Q: How do I determine if a function is one-to-one or onto?

A: To determine if a function is one-to-one or onto, you need to check the following conditions:

• One-to-one: A function is one-to-one if it assigns each input to a unique output. This means that if $f(a) = f(b)$, then $a = b$.
• Onto: A function is onto if it assigns each output to at least one input. This means that for every output $y$, there exists an input $x$ such that $f(x) = y$.

Q: Can I always find a function that is the composition of two other functions?

A: No, not always. In some cases, it may not be possible to find a function that is the composition of two other functions. This is because the composition of functions may not exist or may not be unique.

Q: How do I determine if a function is a bijection?

A: To determine if a function is a bijection, you need to check the following conditions:

• One-to-one: A function is one-to-one if it assigns each input to a unique output. This means that if $f(a) = f(b)$, then $a = b$.
• Onto: A function is onto if it assigns each output to at least one input. This means that for every output $y$, there exists an input $x$ such that $f(x) = y$.

Q: Can I always find a function that is the inverse of a bijection?

A: Yes, if a function is a bijection, then it has an inverse function. This is because a bijection is both one-to-one and onto, which means that it has a unique inverse function.

Conclusion

In conclusion, function composition is an important concept in mathematics that allows us to create new functions from existing functions. By understanding function composition, we can gain insights into the behavior of complex systems and make informed decisions. We hope that this article has answered some of your frequently asked questions about function composition.