Select The Correct Answer.Consider These Functions:$\[ \begin{array}{l} f(x)=-\frac{1}{2} X^2+5x \\ g(x)=x^2+2 \end{array} \\]What Is The Value Of \[$ F(g(-2)) \$\]?A. -28 B. -12 C. 12 D. 146

Introduction to Composite Functions

In mathematics, a composite function is a function that is derived from two or more functions. It is a way of combining functions to create a new function. Composite functions are used to solve problems that involve multiple functions and are an essential concept in calculus and other branches of mathematics.

Understanding the Given Functions

We are given two functions:

• $f(x)=-\frac{1}{2} x^2+5x$
• $g(x)=x^2+2$

These functions are defined as follows:

• The function $f(x)$ is a quadratic function with a negative coefficient of $x^2$, indicating a downward-opening parabola. The function has a maximum value at $x=0$.
• The function $g(x)$ is a quadratic function with a positive coefficient of $x^2$, indicating an upward-opening parabola.

Evaluating the Composite Function

To evaluate the composite function $f(g(-2))$, we need to follow the order of operations:

1. Evaluate the inner function $g(-2)$.
2. Substitute the result into the outer function $f(x)$.

Step 1: Evaluate the Inner Function

To evaluate the inner function $g(-2)$, we substitute $x=-2$ into the function $g(x)=x^2+2$.

$g(-2)=(-2)^2+2$

Step 2: Simplify the Expression

Simplifying the expression, we get:

$g(-2)=4+2$

$g(-2)=6$

Step 3: Substitute the Result into the Outer Function

Now that we have evaluated the inner function, we substitute the result into the outer function $f(x)=-\frac{1}{2} x^2+5x$.

$f(g(-2))=f(6)$

Step 4: Evaluate the Outer Function

To evaluate the outer function $f(6)$, we substitute $x=6$ into the function $f(x)=-\frac{1}{2} x^2+5x$.

$f(6)=-\frac{1}{2} (6)^2+5(6)$

Step 5: Simplify the Expression

Simplifying the expression, we get:

$f(6)=-\frac{1}{2} (36)+30$

$f(6)=-18+30$

$f(6)=12$

Conclusion

Therefore, the value of $f(g(-2))$ is $\boxed{12}$.

Discussion

The concept of composite functions is essential in mathematics, particularly in calculus. It allows us to solve problems that involve multiple functions and is used extensively in physics, engineering, and other fields. In this problem, we evaluated the composite function $f(g(-2))$ by following the order of operations and simplifying the expressions. The result is a value of 12, which is the correct answer.

Final Answer

The final answer is $\boxed{12}$.

Introduction

In the previous article, we evaluated the composite function $f(g(-2))$ using the given functions $f(x)=-\frac{1}{2} x^2+5x$ and $g(x)=x^2+2$. In this article, we will answer some frequently asked questions related to composite functions and provide additional examples to help solidify the concept.

Q1: What is a composite function?

A composite function is a function that is derived from two or more functions. It is a way of combining functions to create a new function.

Q2: How do I evaluate a composite function?

To evaluate a composite function, you need to follow the order of operations:

1. Evaluate the inner function.
2. Substitute the result into the outer function.
3. Simplify the expression.

Q3: What is the difference between a composite function and a function of a function?

A composite function is a function that is derived from two or more functions, whereas a function of a function is a function that takes another function as its input.

Q4: Can I use composite functions to solve real-world problems?

Yes, composite functions are used extensively in physics, engineering, and other fields to solve problems that involve multiple functions.

Q5: How do I know which function to evaluate first in a composite function?

You should always evaluate the inner function first, as it is the function that is being substituted into the outer function.

Q6: Can I use composite functions with different types of functions?

Yes, composite functions can be used with different types of functions, such as linear, quadratic, polynomial, and trigonometric functions.

Q7: How do I simplify a composite function?

To simplify a composite function, you need to follow the order of operations and simplify the expressions.

Q8: Can I use composite functions to solve optimization problems?

Yes, composite functions can be used to solve optimization problems by finding the maximum or minimum value of a function.

Q9: How do I know if a composite function is increasing or decreasing?

You can determine if a composite function is increasing or decreasing by evaluating the derivative of the function.

Q10: Can I use composite functions to solve problems in calculus?

Yes, composite functions are used extensively in calculus to solve problems that involve multiple functions.

Additional Examples

Example 1: Evaluating a Composite Function

Evaluate the composite function $f(g(3))$ using the given functions $f(x)=x^2+2x$ and $g(x)=x-1$.

Solution

To evaluate the composite function $f(g(3))$, we need to follow the order of operations:

1. Evaluate the inner function $g(3)$.
2. Substitute the result into the outer function $f(x)$.

$g(3)=(3)-1$

$g(3)=2$

$f(g(3))=f(2)$

$f(2)=(2)^2+2(2)$

$f(2)=4+4$

$f(2)=8$

Therefore, the value of $f(g(3))$ is $\boxed{8}$.

Example 2: Evaluating a Composite Function with Different Types of Functions

Evaluate the composite function $f(g(x))$ using the given functions $f(x)=\sin x$ and $g(x)=x^2+1$.

Solution

To evaluate the composite function $f(g(x))$, we need to follow the order of operations:

1. Evaluate the inner function $g(x)$.
2. Substitute the result into the outer function $f(x)$.

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

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

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

Therefore, the value of $f(g(x))$ is $\boxed{\sin (x^2+1)}$.

Conclusion

In this article, we answered some frequently asked questions related to composite functions and provided additional examples to help solidify the concept. We also evaluated the composite function $f(g(3))$ using the given functions $f(x)=x^2+2x$ and $g(x)=x-1$, and the composite function $f(g(x))$ using the given functions $f(x)=\sin x$ and $g(x)=x^2+1$. The results are $\boxed{8}$ and $\boxed{\sin (x^2+1)}$, respectively.

Final Answer

The final answer is $\boxed{8}$ and $\boxed{\sin (x^2+1)}$.