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

Introduction

In mathematics, combining functions is a fundamental concept that allows us to create new functions by combining existing ones. In this article, we will explore the process of evaluating combined functions and matching them to their corresponding categories. We will use two given functions, $f(x)$ and $g(x)$, to demonstrate this concept.

Function Definitions

Before we begin, let's define the two given functions:

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

This is a quadratic function that takes an input $x$ and returns the value $1 - x^2$. The graph of this function is a parabola that opens downwards.

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

This is a square root function that takes an input $x$ and returns the value $\sqrt{11 - 4x}$. The graph of this function is a curve that opens upwards.

Evaluating Combined Functions

Now that we have defined the two functions, let's evaluate some combined functions by plugging in different values of $x$.

$f(g(x))$

To evaluate this combined function, we need to plug in the value of $g(x)$ into the function $f(x)$. This gives us:

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

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

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

Simplifying this expression, we get:

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

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

This is the final expression for the combined function $f(g(x))$.

$g(f(x))$

To evaluate this combined function, we need to plug in the value of $f(x)$ into the function $g(x)$. This gives us:

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

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

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

Simplifying this expression, we get:

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

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

This is the final expression for the combined function $g(f(x))$.

Matching Combined Functions to Categories

Now that we have evaluated the combined functions, let's match them to their corresponding categories.

$f(g(x))$

This combined function is a linear function, as it can be written in the form $y = mx + b$, where $m$ and $b$ are constants. Specifically, the function $f(g(x)) = -10 + 4x$ is a linear function with a slope of $4$ and a y-intercept of $-10$.

$g(f(x))$

This combined function is a square root function, as it involves the square root of an expression. Specifically, the function $g(f(x)) = \sqrt{7 + 4x^2}$ is a square root function with a domain of $x \geq -\sqrt{7/4}$.

Conclusion

In this article, we have explored the process of evaluating combined functions and matching them to their corresponding categories. We have used two given functions, $f(x)$ and $g(x)$, to demonstrate this concept. By evaluating the combined functions $f(g(x))$ and $g(f(x))$, we have shown that they can be matched to their corresponding categories: linear functions and square root functions, respectively.

Discussion

• What are some other examples of combined functions that can be evaluated and matched to their corresponding categories?
• How can we use combined functions to model real-world phenomena?
• What are some potential applications of combined functions in mathematics and other fields?

References

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

Glossary

• Combined function: A function that is created by combining two or more existing functions.
• Linear function: A function that can be written in the form $y = mx + b$, where $m$ and $b$ are constants.
• Square root function: A function that involves the square root of an expression.
  Evaluating Combined Functions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of evaluating combined functions and matching them to their corresponding categories. In this article, we will answer some frequently asked questions about combined functions and provide additional guidance on how to evaluate and match them.

Q&A

Q: What is a combined function?

A: A combined function is a function that is created by combining two or more existing functions. For example, if we have two functions $f(x)$ and $g(x)$, we can create a new function $f(g(x))$ by plugging in the value of $g(x)$ into the function $f(x)$.

Q: How do I evaluate a combined function?

A: To evaluate a combined function, you need to follow the order of operations (PEMDAS):

1. Evaluate the inner function (in this case, $g(x)$).
2. Plug the value of the inner function into the outer function (in this case, $f(x)$).
3. Simplify the resulting expression.

Q: What are some common types of combined functions?

A: Some common types of combined functions include:

• Linear functions: These are functions that can be written in the form $y = mx + b$, where $m$ and $b$ are constants.
• Square root functions: These are functions that involve the square root of an expression.
• Polynomial functions: These are functions that involve the sum of multiple terms, each of which is a power of $x$.

Q: How do I match a combined function to its corresponding category?

A: To match a combined function to its corresponding category, you need to analyze the function and determine its type. For example, if a combined function involves the square root of an expression, it is likely a square root function.

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

A: Combined functions have many real-world applications, including:

• Modeling population growth: Combined functions can be used to model the growth of a population over time.
• Analyzing financial data: Combined functions can be used to analyze financial data and make predictions about future trends.
• Designing electrical circuits: Combined functions can be used to design electrical circuits and predict their behavior.

Q: How can I use combined functions to solve problems?

A: To use combined functions to solve problems, you need to:

1. Identify the problem and determine the type of function that is needed to solve it.
2. Create a combined function that meets the requirements of the problem.
3. Evaluate the combined function and use the results to solve the problem.

Examples

Example 1: Evaluating a Combined Function

Suppose we have two functions $f(x) = 1 - x^2$ and $g(x) = \sqrt{11 - 4x}$. We want to evaluate the combined function $f(g(x))$.

To evaluate this function, we need to follow the order of operations:

1. Evaluate the inner function $g(x)$: $g(x) = \sqrt{11 - 4x}$.
2. Plug the value of the inner function into the outer function $f(x)$: $f(g(x)) = 1 - (g(x))^2$.
3. Simplify the resulting expression: $f(g(x)) = 1 - (\sqrt{11 - 4x})^2 = 1 - (11 - 4x) = -10 + 4x$.

Example 2: Matching a Combined Function to its Corresponding Category

Suppose we have a combined function $h(x) = \sqrt{7 + 4x^2}$. We want to match this function to its corresponding category.

To match this function to its corresponding category, we need to analyze the function and determine its type. In this case, the function involves the square root of an expression, so it is likely a square root function.

Conclusion

In this article, we have answered some frequently asked questions about combined functions and provided additional guidance on how to evaluate and match them. We have also provided examples of how to use combined functions to solve problems and match them to their corresponding categories.

Discussion

• What are some other types of combined functions that can be evaluated and matched to their corresponding categories?
• How can we use combined functions to model real-world phenomena?
• What are some potential applications of combined functions in mathematics and other fields?

References

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

Glossary

• Combined function: A function that is created by combining two or more existing functions.
• Linear function: A function that can be written in the form $y = mx + b$, where $m$ and $b$ are constants.
• Square root function: A function that involves the square root of an expression.