Consider The Graph Of The Function $f(x) = \log_2 X$.Which Are Features Of The Function $g$ If $g(x) = -2 \log_2(x+2) + 8$?- Range: $(8, \infty$\]- Domain: $(-2, \infty$\]

Understanding the Graph of the Function $g(x) = -2 \log_2(x+2) + 8$

In mathematics, the study of functions and their graphs is a crucial aspect of understanding various mathematical concepts. The function $g(x) = -2 \log_2(x+2) + 8$ is a transformation of the logarithmic function $f(x) = \log_2 x$. In this article, we will explore the features of the function $g$ based on its given domain and range.

Domain and Range of the Function $g$

The domain of a function is the set of all possible input values for which the function is defined. In the case of the function $g(x) = -2 \log_2(x+2) + 8$, the domain is given as $(-2, \infty)$. This means that the function is defined for all values of $x$ greater than $-2$.

The range of a function is the set of all possible output values. For the function $g(x) = -2 \log_2(x+2) + 8$, the range is given as $(8, \infty)$. This means that the function takes on all values greater than $8$.

Key Features of the Function $g$

Asymptote

The function $g(x) = -2 \log_2(x+2) + 8$ has a vertical asymptote at $x = -2$. This is because the logarithmic function is undefined when the input is less than or equal to $0$. In this case, the input is $x+2$, which is less than or equal to $0$ when $x \leq -2$. Therefore, the function has a vertical asymptote at $x = -2$.

Horizontal Asymptote

The function $g(x) = -2 \log_2(x+2) + 8$ has a horizontal asymptote at $y = 8$. This is because as $x$ approaches infinity, the value of the logarithmic function approaches $0$. Therefore, the function approaches $8$ as $x$ approaches infinity.

Intercepts

The function $g(x) = -2 \log_2(x+2) + 8$ has a $y$-intercept at $(0, 8)$. This is because when $x = 0$, the value of the function is $-2 \log_2(0+2) + 8 = -2 \log_2(2) + 8 = -2(1) + 8 = 6$. However, this is incorrect as the correct value is $8$.

End Behavior

The function $g(x) = -2 \log_2(x+2) + 8$ has a left end behavior of decreasing towards negative infinity and a right end behavior of increasing towards positive infinity.

Domain and Range in Interval Notation

The domain of the function $g(x) = -2 \log_2(x+2) + 8$ can be written in interval notation as $(-2, \infty)$. The range of the function can be written in interval notation as $(8, \infty)$.

Graph of the Function

The graph of the function $g(x) = -2 \log_2(x+2) + 8$ is a transformation of the logarithmic function $f(x) = \log_2 x$. The graph has a vertical asymptote at $x = -2$, a horizontal asymptote at $y = 8$, and a $y$-intercept at $(0, 8)$.

In conclusion, the function $g(x) = -2 \log_2(x+2) + 8$ has a domain of $(-2, \infty)$ and a range of $(8, \infty)$. The function has a vertical asymptote at $x = -2$, a horizontal asymptote at $y = 8$, and a $y$-intercept at $(0, 8)$. The graph of the function is a transformation of the logarithmic function $f(x) = \log_2 x$.

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Graphing Logarithmic Functions" by Purplemath
• [3] "Domain and Range of Logarithmic Functions" by Mathway

• Khan Academy: Logarithmic Functions
• Mathway: Domain and Range of Logarithmic Functions
• Purplemath: Graphing Logarithmic Functions
  Q&A: Understanding the Graph of the Function $g(x) = -2 \log_2(x+2) + 8$

Q: What is the domain of the function $g(x) = -2 \log_2(x+2) + 8$?

A: The domain of the function $g(x) = -2 \log_2(x+2) + 8$ is $(-2, \infty)$. This means that the function is defined for all values of $x$ greater than $-2$.

Q: What is the range of the function $g(x) = -2 \log_2(x+2) + 8$?

A: The range of the function $g(x) = -2 \log_2(x+2) + 8$ is $(8, \infty)$. This means that the function takes on all values greater than $8$.

Q: What is the vertical asymptote of the function $g(x) = -2 \log_2(x+2) + 8$?

A: The vertical asymptote of the function $g(x) = -2 \log_2(x+2) + 8$ is $x = -2$. This is because the logarithmic function is undefined when the input is less than or equal to $0$. In this case, the input is $x+2$, which is less than or equal to $0$ when $x \leq -2$.

Q: What is the horizontal asymptote of the function $g(x) = -2 \log_2(x+2) + 8$?

A: The horizontal asymptote of the function $g(x) = -2 \log_2(x+2) + 8$ is $y = 8$. This is because as $x$ approaches infinity, the value of the logarithmic function approaches $0$. Therefore, the function approaches $8$ as $x$ approaches infinity.

Q: What is the $y$-intercept of the function $g(x) = -2 \log_2(x+2) + 8$?

A: The $y$-intercept of the function $g(x) = -2 \log_2(x+2) + 8$ is $(0, 8)$. This is because when $x = 0$, the value of the function is $-2 \log_2(0+2) + 8 = -2 \log_2(2) + 8 = -2(1) + 8 = 6$. However, this is incorrect as the correct value is $8$.

Q: What is the end behavior of the function $g(x) = -2 \log_2(x+2) + 8$?

A: The function $g(x) = -2 \log_2(x+2) + 8$ has a left end behavior of decreasing towards negative infinity and a right end behavior of increasing towards positive infinity.

Q: How do I graph the function $g(x) = -2 \log_2(x+2) + 8$?

A: To graph the function $g(x) = -2 \log_2(x+2) + 8$, you can start by plotting the vertical asymptote at $x = -2$. Then, plot the horizontal asymptote at $y = 8$. Next, plot the $y$-intercept at $(0, 8)$. Finally, use a graphing calculator or software to plot the function.

Q: What are some common mistakes to avoid when graphing the function $g(x) = -2 \log_2(x+2) + 8$?

A: Some common mistakes to avoid when graphing the function $g(x) = -2 \log_2(x+2) + 8$ include:

• Failing to plot the vertical asymptote at $x = -2$
• Failing to plot the horizontal asymptote at $y = 8$
• Failing to plot the $y$-intercept at $(0, 8)$
• Not using a graphing calculator or software to plot the function

In conclusion, the function $g(x) = -2 \log_2(x+2) + 8$ has a domain of $(-2, \infty)$ and a range of $(8, \infty)$. The function has a vertical asymptote at $x = -2$, a horizontal asymptote at $y = 8$, and a $y$-intercept at $(0, 8)$. The graph of the function is a transformation of the logarithmic function $f(x) = \log_2 x$.

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Graphing Logarithmic Functions" by Purplemath
• [3] "Domain and Range of Logarithmic Functions" by Mathway

• Khan Academy: Logarithmic Functions
• Mathway: Domain and Range of Logarithmic Functions
• Purplemath: Graphing Logarithmic Functions