Given The Function \[$ F \$\] Is Even, And The Function \[$ G \$\] Is Odd, Determine The Values Of \[$ A \$\] And \[$ B \$\].For Function \[$ F \$\]:$\[ \begin{array}{|c|c|} \hline x & F(x) \\ \hline -2 & 4

Determining the Values of a and b for Even and Odd Functions

In mathematics, functions can be classified into even and odd functions based on their symmetry properties. An even function satisfies the condition $f(-x) = f(x)$ for all $x$ in its domain, while an odd function satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. In this article, we will discuss how to determine the values of $a$ and $b$ for a given even function $f$ and an odd function $g$.

An even function is a function that remains unchanged when its input is negated. Mathematically, this can be expressed as $f(-x) = f(x)$ for all $x$ in the domain of $f$. Some common examples of even functions include:

• $f(x) = x^2$
• $f(x) = \cos x$
• $f(x) = |x|$

On the other hand, an odd function is a function that changes sign when its input is negated. Mathematically, this can be expressed as $f(-x) = -f(x)$ for all $x$ in the domain of $f$. Some common examples of odd functions include:

• $f(x) = x^3$
• $f(x) = \sin x$
• $f(x) = x$

We are given two functions, $f$ and $g$, where $f$ is an even function and $g$ is an odd function. The function $f$ is defined as:

$$f(x) = \begin{cases} 4 & \text{if } x = -2 \\ a & \text{if } x = 2 \\ b & \text{if } x = 4 \end{cases}$$

The function $g$ is defined as:

$$g(x) = \begin{cases} -1 & \text{if } x = -2 \\ a & \text{if } x = 2 \\ b & \text{if } x = 4 \end{cases}$$

Since $f$ is an even function, we know that $f(-x) = f(x)$ for all $x$ in the domain of $f$. Therefore, we can write:

$$f(-2) = f(2) = a$$

$$f(-4) = f(4) = b$$

Since $f(-2) = 4$, we can conclude that $a = 4$. Similarly, since $f(4) = b$, we can conclude that $b = 4$.

On the other hand, since $g$ is an odd function, we know that $g(-x) = -g(x)$ for all $x$ in the domain of $g$. Therefore, we can write:

$$g(-2) = -g(2) = -a$$

$$g(-4) = -g(4) = -b$$

Since $g(-2) = -1$, we can conclude that $-a = -1$, which implies that $a = 1$. Similarly, since $g(-4) = -b$, we can conclude that $-b = -1$, which implies that $b = 1$.

In conclusion, we have determined the values of $a$ and $b$ for the given even function $f$ and odd function $g$. We found that $a = 4$ and $b = 4$ for the even function $f$, and $a = 1$ and $b = 1$ for the odd function $g$.

• [1] "Even and Odd Functions" by Math Open Reference
• [2] "Symmetry of Functions" by Wolfram MathWorld

This problem is a classic example of how to determine the values of $a$ and $b$ for a given even function and an odd function. The key concept here is the symmetry property of even and odd functions, which allows us to write the function values in terms of $a$ and $b$. We hope that this article has provided a clear and concise explanation of how to solve this problem.

• [1] "Even and Odd Functions" by Khan Academy
• [2] "Symmetry of Functions" by MIT OpenCourseWare

Q: What is the difference between an even function and an odd function?

A: An even function is a function that remains unchanged when its input is negated, i.e., $f(-x) = f(x)$ for all $x$ in the domain of $f$. On the other hand, an odd function is a function that changes sign when its input is negated, i.e., $f(-x) = -f(x)$ for all $x$ in the domain of $f$.

Q: Can you give some examples of even functions?

A: Yes, some common examples of even functions include:

• $f(x) = x^2$
• $f(x) = \cos x$
• $f(x) = |x|$

Q: Can you give some examples of odd functions?

A: Yes, some common examples of odd functions include:

• $f(x) = x^3$
• $f(x) = \sin x$
• $f(x) = x$

Q: How do you determine if a function is even or odd?

A: To determine if a function is even or odd, you can use the following criteria:

• If $f(-x) = f(x)$ for all $x$ in the domain of $f$, then $f$ is an even function.
• If $f(-x) = -f(x)$ for all $x$ in the domain of $f$, then $f$ is an odd function.

Q: What is the significance of even and odd functions in mathematics?

A: Even and odd functions play a significant role in mathematics, particularly in the study of symmetry and periodicity. They are used to describe the behavior of functions and to identify patterns and relationships between different mathematical objects.

Q: Can you give an example of how to use even and odd functions in real-world applications?

A: Yes, even and odd functions are used in a variety of real-world applications, including:

• Physics: Even and odd functions are used to describe the behavior of waves and oscillations.
• Engineering: Even and odd functions are used to design and analyze systems that involve symmetry and periodicity.
• Computer Science: Even and odd functions are used in algorithms and data structures to identify patterns and relationships between different data points.

Q: How do you determine the values of a and b for a given even function f and an odd function g?

A: To determine the values of a and b for a given even function f and an odd function g, you can use the following steps:

• For the even function f, use the fact that $f(-x) = f(x)$ for all $x$ in the domain of f to write the function values in terms of a and b.
• For the odd function g, use the fact that $g(-x) = -g(x)$ for all $x$ in the domain of g to write the function values in terms of a and b.
• Solve the resulting equations to determine the values of a and b.

Q: Can you give an example of how to determine the values of a and b for a given even function f and an odd function g?

A: Yes, consider the following example:

$$f(x) = \begin{cases} 4 & \text{if } x = -2 \\ a & \text{if } x = 2 \\ b & \text{if } x = 4 \end{cases}$$

$$g(x) = \begin{cases} -1 & \text{if } x = -2 \\ a & \text{if } x = 2 \\ b & \text{if } x = 4 \end{cases}$$

Using the fact that $f(-x) = f(x)$ for all $x$ in the domain of f, we can write:

$$f(-2) = f(2) = a$$

$$f(-4) = f(4) = b$$

Since $f(-2) = 4$, we can conclude that $a = 4$. Similarly, since $f(4) = b$, we can conclude that $b = 4$.

Using the fact that $g(-x) = -g(x)$ for all $x$ in the domain of g, we can write:

$$g(-2) = -g(2) = -a$$

$$g(-4) = -g(4) = -b$$

Since $g(-2) = -1$, we can conclude that $-a = -1$, which implies that $a = 1$. Similarly, since $g(-4) = -b$, we can conclude that $-b = -1$, which implies that $b = 1$.

Therefore, the values of a and b are $a = 4$ and $b = 4$ for the even function f, and $a = 1$ and $b = 1$ for the odd function g.

In conclusion, even and odd functions are an important concept in mathematics, and they have many real-world applications. By understanding the properties and behavior of even and odd functions, we can solve problems and make predictions in a variety of fields. We hope that this Q&A article has provided a clear and concise explanation of even and odd functions and their applications.