The Function F F F Is Defined By F ( X ) = ( X − 6 ) ( X − 2 ) ( X + 6 F(x) = (x - 6)(x - 2)(x + 6 F ( X ) = ( X − 6 ) ( X − 2 ) ( X + 6 ]. In The X Y Xy X Y -plane, The Graph Of Y = G ( X Y = G(x Y = G ( X ] Is The Result Of Translating The Graph Of Y = F ( X Y = F(x Y = F ( X ] Up 4 Units. What Is The Value Of G ( 0 G(0 G ( 0 ]?

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Introduction

In mathematics, functions and their graphs are essential concepts that help us understand various mathematical relationships. The function $f$ defined by $f(x) = (x - 6)(x - 2)(x + 6)$ is a polynomial function that can be graphed in the $xy$-plane. However, when we are given a new function $g(x)$, which is the result of translating the graph of $f(x)$ up 4 units, we need to understand how this translation affects the graph of $f(x)$ and find the value of $g(0)$.

Understanding the Graph of $f(x)$

The graph of $f(x) = (x - 6)(x - 2)(x + 6)$ is a cubic function that has three roots at $x = 6, 2,$ and $-6$. These roots are the points where the graph of $f(x)$ intersects the $x$-axis. The graph of $f(x)$ is a cubic curve that opens upward, meaning that it increases as $x$ increases.

Translating the Graph of $f(x)$

When we translate the graph of $f(x)$ up 4 units, we are essentially shifting the graph of $f(x)$ 4 units upward. This means that for every point $(x, y)$ on the graph of $f(x)$, the corresponding point on the graph of $g(x)$ is $(x, y + 4)$. This translation affects the graph of $f(x)$ by moving it upward, resulting in a new graph $g(x)$.

Finding the Value of $g(0)$

To find the value of $g(0)$, we need to understand how the translation affects the graph of $f(x)$ at the point $x = 0$. Since the graph of $f(x)$ is translated up 4 units, the value of $g(0)$ will be the value of $f(0)$ plus 4.

Calculating the Value of $f(0)$

To calculate the value of $f(0)$, we substitute $x = 0$ into the equation $f(x) = (x - 6)(x - 2)(x + 6)$. This gives us:

$f(0) = (0 - 6)(0 - 2)(0 + 6)$ $f(0) = (-6)(-2)(6)$ $f(0) = 72$

Finding the Value of $g(0)$

Now that we have calculated the value of $f(0)$, we can find the value of $g(0)$ by adding 4 to the value of $f(0)$. This gives us:

$g(0) = f(0) + 4$ $g(0) = 72 + 4$ $g(0) = 76$

Conclusion

In this article, we have understood the function $f$ defined by $f(x) = (x - 6)(x - 2)(x + 6)$ and its translation to form the graph of $g(x)$. We have also calculated the value of $g(0)$ by adding 4 to the value of $f(0)$. The value of $g(0)$ is 76.

Final Answer

The final answer is $\boxed{76}$.

Discussion

The discussion of this problem involves understanding the concept of function translation and its effect on the graph of a function. The translation of the graph of $f(x)$ up 4 units results in a new graph $g(x)$, and we need to find the value of $g(0)$ by understanding how this translation affects the graph of $f(x)$ at the point $x = 0$. This problem requires a good understanding of function notation and graphing concepts.

Related Problems

• Find the value of $f(-2)$.
• Find the value of $g(-2)$.
• Graph the function $f(x) = (x - 6)(x - 2)(x + 6)$ and its translation $g(x)$.

Solutions

• To find the value of $f(-2)$, we substitute $x = -2$ into the equation $f(x) = (x - 6)(x - 2)(x + 6)$. This gives us:

$f(-2) = (-2 - 6)(-2 - 2)(-2 + 6)$ $f(-2) = (-8)(-4)(4)$ $f(-2) = 128$

• To find the value of $g(-2)$, we substitute $x = -2$ into the equation $g(x) = f(x) + 4$. This gives us:

$g(-2) = f(-2) + 4$ $g(-2) = 128 + 4$ $g(-2) = 132$

• To graph the function $f(x) = (x - 6)(x - 2)(x + 6)$ and its translation $g(x)$, we can use a graphing calculator or software. The graph of $f(x)$ is a cubic curve that opens upward, and the graph of $g(x)$ is the same curve translated up 4 units.

Conclusion

In this article, we have discussed the function $f$ defined by $f(x) = (x - 6)(x - 2)(x + 6)$ and its translation to form the graph of $g(x)$. We have also calculated the value of $g(0)$ by adding 4 to the value of $f(0)$. The value of $g(0)$ is 76.

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Introduction

In our previous article, we discussed the function $f$ defined by $f(x) = (x - 6)(x - 2)(x + 6)$ and its translation to form the graph of $g(x)$. We also calculated the value of $g(0)$ by adding 4 to the value of $f(0)$. In this article, we will answer some frequently asked questions about the function $f$ and its translation.

Q1: What is the domain of the function $f(x)$?

A1: The domain of the function $f(x)$ is all real numbers, since the function is defined for all values of $x$.

Q2: What is the range of the function $f(x)$?

A2: The range of the function $f(x)$ is all real numbers greater than or equal to 0, since the function is a cubic function that opens upward.

Q3: How does the translation of the graph of $f(x)$ affect the graph of $g(x)$?

A3: The translation of the graph of $f(x)$ up 4 units results in a new graph $g(x)$ that is the same as the graph of $f(x)$ but shifted upward by 4 units.

Q4: How do we find the value of $g(x)$ for any value of $x$?

A4: To find the value of $g(x)$ for any value of $x$, we substitute $x$ into the equation $g(x) = f(x) + 4$.

Q5: What is the value of $g(0)$?

A5: The value of $g(0)$ is 76, since we calculated $g(0) = f(0) + 4$ and $f(0) = 72$.

Q6: How do we graph the function $f(x)$ and its translation $g(x)$?

A6: We can use a graphing calculator or software to graph the function $f(x)$ and its translation $g(x)$. The graph of $f(x)$ is a cubic curve that opens upward, and the graph of $g(x)$ is the same curve translated up 4 units.

Q7: What is the relationship between the function $f(x)$ and its translation $g(x)$?

A7: The function $f(x)$ and its translation $g(x)$ are related by the equation $g(x) = f(x) + 4$, which means that the graph of $g(x)$ is the same as the graph of $f(x)$ but shifted upward by 4 units.

Q8: How do we find the inverse of the function $f(x)$?

A8: To find the inverse of the function $f(x)$, we need to solve the equation $y = (x - 6)(x - 2)(x + 6)$ for $x$ in terms of $y$. This will give us the inverse function $f^{-1}(x)$.

Q9: What is the value of $f^{-1}(0)$?

A9: To find the value of $f^{-1}(0)$, we need to solve the equation $y = (x - 6)(x - 2)(x + 6)$ for $x$ when $y = 0$. This will give us the value of $x$ that corresponds to $y = 0$.

Q10: How do we graph the inverse of the function $f(x)$?

A10: We can use a graphing calculator or software to graph the inverse of the function $f(x)$. The graph of the inverse function $f^{-1}(x)$ is a reflection of the graph of $f(x)$ across the line $y = x$.

Conclusion

In this article, we have answered some frequently asked questions about the function $f$ and its translation. We have discussed the domain and range of the function $f(x)$, the effect of the translation on the graph of $g(x)$, and how to find the value of $g(x)$ for any value of $x$. We have also discussed the relationship between the function $f(x)$ and its translation $g(x)$, and how to find the inverse of the function $f(x)$.