Drag The Tiles To The Boxes To Form Correct Pairs.Consider Functions $f$ And $g$:$\begin{array}{l} f(x)=1-x^2 \\ g(x)=\sqrt{11-4x} \end{array}$Evaluate Each Combined Function, And Match It To The Corresponding Value:1.

Introduction

In mathematics, combining functions is a fundamental concept that allows us to create new functions by applying existing ones. In this article, we will explore two given functions, $f(x)$ and $g(x)$, and evaluate their combined forms. We will then match each combined function to its corresponding value, providing a deeper understanding of function composition and its applications.

Function $f(x)$

The first function, $f(x)$, is defined as:

$$f(x) = 1 - x^2$$

This is a quadratic function, which means it has a parabolic shape. The graph of $f(x)$ opens downwards, indicating that the function decreases as $x$ increases.

Function $g(x)$

The second function, $g(x)$, is defined as:

$$g(x) = \sqrt{11 - 4x}$$

This is a square root function, which means it has a restricted domain. The expression inside the square root must be non-negative, so we have the inequality $11 - 4x \geq 0$. Solving for $x$, we get $x \leq \frac{11}{4}$.

Combined Functions

Now that we have defined $f(x)$ and $g(x)$, we can combine them to create new functions. There are two possible combinations:

1. $f(g(x))$: This is the composition of $f(x)$ and $g(x)$, where the output of $g(x)$ is used as the input for $f(x)$.
2. $g(f(x))$: This is the composition of $g(x)$ and $f(x)$, where the output of $f(x)$ is used as the input for $g(x)$.

Evaluating $f(g(x))$

To evaluate $f(g(x))$, we need to substitute $g(x)$ into $f(x)$:

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

Using the definition of $g(x)$, we get:

$$f(g(x)) = 1 - (\sqrt{11 - 4x})^2$$

Simplifying the expression, we get:

$$f(g(x)) = 1 - (11 - 4x)$$

$$f(g(x)) = -10 + 4x$$

Evaluating $g(f(x))$

To evaluate $g(f(x))$, we need to substitute $f(x)$ into $g(x)$:

$$g(f(x)) = \sqrt{11 - 4f(x)}$$

Using the definition of $f(x)$, we get:

$$g(f(x)) = \sqrt{11 - 4(1 - x^2)}$$

Simplifying the expression, we get:

$$g(f(x)) = \sqrt{11 - 4 + 4x^2}$$

$$g(f(x)) = \sqrt{7 + 4x^2}$$

Matching Combined Functions to Values

Now that we have evaluated the combined functions, we need to match them to their corresponding values. Let's consider the following values:

Value	Combined Function
1	$f(g(x))$
2	$g(f(x))$

Based on our evaluations, we can see that:

• $f(g(x)) = -10 + 4x$
• $g(f(x)) = \sqrt{7 + 4x^2}$

Therefore, we can match the combined functions to their corresponding values as follows:

Value	Combined Function
1	$-10 + 4x$
2	$\sqrt{7 + 4x^2}$

Conclusion

In this article, we have evaluated two combined functions, $f(g(x))$ and $g(f(x))$, and matched them to their corresponding values. We have seen that the combined functions have different forms and behaviors, depending on the order in which the functions are composed. This highlights the importance of understanding function composition and its applications in mathematics.

Discussion

• What are some real-world applications of function composition?
• How do you think the combined functions would behave if the input values were restricted to a specific range?
• Can you think of any other ways to combine the functions $f(x)$ and $g(x)$?

References

• [1] "Function Composition" by Khan Academy
• [2] "Quadratic Functions" by Math Open Reference
• [3] "Square Root Functions" by Purplemath
  Evaluating Combined Functions: A Mathematical Puzzle - Q&A ===========================================================

Introduction

In our previous article, we explored the concept of combined functions and evaluated two specific functions, $f(x)$ and $g(x)$. We also matched the combined functions to their corresponding values. In this article, we will answer some frequently asked questions (FAQs) related to combined functions and provide additional insights into this fascinating topic.

Q&A

Q: What is function composition?

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

Q: What are the two combined functions we evaluated in the previous article?

A: The two combined functions we evaluated were $f(g(x))$ and $g(f(x))$. $f(g(x))$ is the composition of $f(x)$ and $g(x)$, where the output of $g(x)$ is used as the input for $f(x)$. $g(f(x))$ is the composition of $g(x)$ and $f(x)$, where the output of $f(x)$ is used as the input for $g(x)$.

Q: How do you evaluate a combined function?

A: To evaluate a combined function, you need to substitute the output of one function into the other function. For example, to evaluate $f(g(x))$, you would substitute $g(x)$ into $f(x)$.

Q: What are some real-world applications of function composition?

A: Function composition has many real-world applications, including:

• Computer graphics: Function composition is used to create complex graphics and animations.
• Signal processing: Function composition is used to process and analyze signals in various fields, such as audio and image processing.
• Optimization: Function composition is used to optimize complex systems and processes.

Q: How do you match combined functions to their corresponding values?

A: To match combined functions to their corresponding values, you need to evaluate the combined function and compare it to the given value. In our previous article, we matched the combined functions to their corresponding values as follows:

Value	Combined Function
1	$-10 + 4x$
2	$\sqrt{7 + 4x^2}$

Q: Can you think of any other ways to combine the functions $f(x)$ and $g(x)$?

A: Yes, there are many other ways to combine the functions $f(x)$ and $g(x)$. Some examples include:

• $f(f(x))$: This is the composition of $f(x)$ with itself.
• $g(g(x))$: This is the composition of $g(x)$ with itself.
• $f(g(g(x)))$: This is the composition of $f(x)$, $g(x)$, and $g(x)$.

Q: How do you think the combined functions would behave if the input values were restricted to a specific range?

A: If the input values were restricted to a specific range, the combined functions would behave differently. For example, if the input values were restricted to the range $x \in [0, 1]$, the combined functions would be restricted to that range as well.

Conclusion

In this article, we have answered some frequently asked questions related to combined functions and provided additional insights into this fascinating topic. We have seen that function composition has many real-world applications and that there are many ways to combine functions. We hope this article has been helpful in understanding combined functions and their applications.

Discussion

• What are some other real-world applications of function composition?
• How do you think the combined functions would behave if the input values were restricted to a specific range?
• Can you think of any other ways to combine the functions $f(x)$ and $g(x)$?

References

• [1] "Function Composition" by Khan Academy
• [2] "Quadratic Functions" by Math Open Reference
• [3] "Square Root Functions" by Purplemath