Let F ( X ) = 1 3 X − 1 F(x) = \frac{1}{3} \sqrt{x-1} F ( X ) = 3 1 ​ X − 1 ​ And G ( X ) = − F ( X ) + 9 G(x) = -f(x) + 9 G ( X ) = − F ( X ) + 9 .Write A Rule For G ( X G(x G ( X ].$g(x) = $

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Understanding the Functions $f(x)$ and $g(x)$

In this problem, we are given two functions, $f(x)$ and $g(x)$, and we need to find a rule for $g(x)$. To do this, we first need to understand the properties of the given functions.

Function $f(x)$

The function $f(x)$ is defined as $f(x) = \frac{1}{3} \sqrt{x-1}$. This function takes an input value $x$, subtracts 1 from it, takes the square root of the result, and then multiplies it by $\frac{1}{3}$.

Function $g(x)$

The function $g(x)$ is defined as $g(x) = -f(x) + 9$. This function takes the output of $f(x)$, negates it, and then adds 9 to it.

Finding the Rule for $g(x)$

To find the rule for $g(x)$, we need to substitute the definition of $f(x)$ into the definition of $g(x)$. This will give us a single expression that defines $g(x)$ in terms of $x$.

Substituting $f(x)$ into $g(x)$

We start by substituting $f(x) = \frac{1}{3} \sqrt{x-1}$ into $g(x) = -f(x) + 9$. This gives us:

$$g(x) = -\left(\frac{1}{3} \sqrt{x-1}\right) + 9$$

Simplifying the Expression

To simplify the expression, we can start by multiplying the term inside the parentheses by $-1$. This gives us:

$$g(x) = -\frac{1}{3} \sqrt{x-1} + 9$$

Combining the Terms

We can combine the two terms by finding a common denominator. In this case, the common denominator is 3. This gives us:

$$g(x) = \frac{-1}{3} \sqrt{x-1} + \frac{27}{3}$$

Simplifying the Expression Further

We can simplify the expression further by combining the two fractions. This gives us:

$$g(x) = \frac{-1}{3} \sqrt{x-1} + \frac{27}{3} = \frac{-1}{3} \sqrt{x-1} + \frac{27}{3} = \frac{-\sqrt{x-1} + 27}{3}$$

Conclusion

In this problem, we were given two functions, $f(x)$ and $g(x)$, and we needed to find a rule for $g(x)$. We started by understanding the properties of the given functions, and then we substituted the definition of $f(x)$ into the definition of $g(x)$. This gave us a single expression that defines $g(x)$ in terms of $x$. We then simplified the expression by combining the terms and finding a common denominator. The final rule for $g(x)$ is:

$$g(x) = \frac{-\sqrt{x-1} + 27}{3}$$

Final Answer

The final answer is $\boxed{\frac{-\sqrt{x-1} + 27}{3}}$.

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Understanding the Functions $f(x)$ and $g(x)$

In this problem, we are given two functions, $f(x)$ and $g(x)$, and we need to find a rule for $g(x)$. To do this, we first need to understand the properties of the given functions.

Function $f(x)$

The function $f(x)$ is defined as $f(x) = \frac{1}{3} \sqrt{x-1}$. This function takes an input value $x$, subtracts 1 from it, takes the square root of the result, and then multiplies it by $\frac{1}{3}$.

Function $g(x)$

The function $g(x)$ is defined as $g(x) = -f(x) + 9$. This function takes the output of $f(x)$, negates it, and then adds 9 to it.

Finding the Rule for $g(x)$

To find the rule for $g(x)$, we need to substitute the definition of $f(x)$ into the definition of $g(x)$. This will give us a single expression that defines $g(x)$ in terms of $x$.

Substituting $f(x)$ into $g(x)$

We start by substituting $f(x) = \frac{1}{3} \sqrt{x-1}$ into $g(x) = -f(x) + 9$. This gives us:

$$g(x) = -\left(\frac{1}{3} \sqrt{x-1}\right) + 9$$

Simplifying the Expression

To simplify the expression, we can start by multiplying the term inside the parentheses by $-1$. This gives us:

$$g(x) = -\frac{1}{3} \sqrt{x-1} + 9$$

Combining the Terms

We can combine the two terms by finding a common denominator. In this case, the common denominator is 3. This gives us:

$$g(x) = \frac{-1}{3} \sqrt{x-1} + \frac{27}{3}$$

Simplifying the Expression Further

We can simplify the expression further by combining the two fractions. This gives us:

$$g(x) = \frac{-1}{3} \sqrt{x-1} + \frac{27}{3} = \frac{-1}{3} \sqrt{x-1} + \frac{27}{3} = \frac{-\sqrt{x-1} + 27}{3}$$

Q&A

Q: What is the rule for $g(x)$?

A: The rule for $g(x)$ is $\frac{-\sqrt{x-1} + 27}{3}$.

Q: What is the domain of the function $f(x)$?

A: The domain of the function $f(x)$ is all real numbers $x$ such that $x \geq 1$.

Q: What is the range of the function $f(x)$?

A: The range of the function $f(x)$ is all real numbers $y$ such that $y \geq 0$.

Q: What is the domain of the function $g(x)$?

A: The domain of the function $g(x)$ is all real numbers $x$ such that $x \geq 1$.

Q: What is the range of the function $g(x)$?

A: The range of the function $g(x)$ is all real numbers $y$ such that $y \leq 9$.

Q: How do we find the rule for $g(x)$?

A: To find the rule for $g(x)$, we need to substitute the definition of $f(x)$ into the definition of $g(x)$.

Q: What is the final answer for the rule of $g(x)$?

A: The final answer for the rule of $g(x)$ is $\boxed{\frac{-\sqrt{x-1} + 27}{3}}$.

Conclusion

In this article, we have discussed the functions $f(x)$ and $g(x)$, and we have found the rule for $g(x)$. We have also answered some common questions related to the functions $f(x)$ and $g(x)$. The final rule for $g(x)$ is $\frac{-\sqrt{x-1} + 27}{3}$.

Final Answer

The final answer is $\boxed{\frac{-\sqrt{x-1} + 27}{3}}$.