Question
Begin by graphing f left parenthesis x right parenthesis equals log Subscript 2 Baseline x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine the given function's domain and range.
g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis
Question content area bottom left
Part 1
Graph the function g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis. Graph the asymptote of g(x) as a dashed line. Use the graphing tool to graph the function.
Part 2
What is the vertical asymptote of g(x)?
enter your response here
(Type an equation.)
Part 3
What is the domain of g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis?
enter your response here
(Simplify your answer. Type your answer in interval notation.)
Part 4
What is the range of g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis?
enter your response here
(Simplify your answer. Type your answer in interval notation.)
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Question content area right
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g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis
Question content area bottom left
Part 1
Graph the function g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis. Graph the asymptote of g(x) as a dashed line. Use the graphing tool to graph the function.
Part 2
What is the vertical asymptote of g(x)?
enter your response here
(Type an equation.)
Part 3
What is the domain of g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis?
enter your response here
(Simplify your answer. Type your answer in interval notation.)
Part 4
What is the range of g left parenthesis x right parenthesis equals log Subscript 2 Baseline left parenthesis x plus 3 right parenthesis?
enter your response here
(Simplify your answer. Type your answer in interval notation.)
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-8
-6
-4
-2
2
4
6
8
x
y
Answers
GPT-4o mini
To analyze the function \( g(x) = \log_2(x + 3) \) using transformations of the base function \( f(x) = \log_2(x) \), let's go through the steps:
### Part 1: Graphing \( g(x) = \log_2(x + 3) \)
1. **Base Function**: The graph of \( f(x) = \log_2(x) \) has a vertical asymptote at \( x = 0 \), passes through the point \( (1, 0) \) (since \( \log_2(1) = 0 \)), and increases to positive infinity as \( x \) approaches infinity.
2. **Transformation**: The function \( g(x) = \log_2(x + 3) \) is obtained by shifting the base function \( \log_2(x) \) to the left by 3 units.
- Thus, the vertical asymptote shifts from \( x = 0 \) to \( x = -3 \).
- The point \( (1, 0) \) of \( f(x) \) becomes the point \( (-2, 0) \) for \( g(x) \) because substituting \( -2 \) into \( g(x) \) gives \( g(-2) = \log_2(1) = 0 \).
Now graph \( g(x) \):
- Draw the vertical dashed line (asymptote) at \( x = -3 \).
- Plot the point \( (-2, 0) \).
- The graph will rise toward positive infinity as \( x \) goes toward infinity and approach negative infinity as \( x \) approaches \( -3 \) from the right.
### Part 2: Vertical Asymptote of \( g(x) \)
The vertical asymptote of \( g(x) \) is given by:
\[
x = -3
\]
### Part 3: Domain of \( g(x) \)
The function \( \log_2(x + 3) \) is defined when the argument \( x + 3 > 0 \):
\[
x + 3 > 0 \implies x > -3
\]
Thus, the domain of \( g(x) \) in interval notation is:
\[
(-3, \infty)
\]
### Part 4: Range of \( g(x) \)
The range of the logarithmic function \( g(x) = \log_2(x + 3) \) is all real numbers, as it can take any value from negative infinity to positive infinity, since the logarithm can grow indefinitely:
\[
(-\infty, \infty)
\]
### Summary of Responses
- **Vertical Asymptote**: \( x = -3 \)
- **Domain**: \( (-3, \infty) \)
- **Range**: \( (-\infty, \infty) \)
This completes the analysis of the function \( g(x) = \log_2(x + 3) \). You can now graph this function based on the information provided.
### Part 1: Graphing \( g(x) = \log_2(x + 3) \)
1. **Base Function**: The graph of \( f(x) = \log_2(x) \) has a vertical asymptote at \( x = 0 \), passes through the point \( (1, 0) \) (since \( \log_2(1) = 0 \)), and increases to positive infinity as \( x \) approaches infinity.
2. **Transformation**: The function \( g(x) = \log_2(x + 3) \) is obtained by shifting the base function \( \log_2(x) \) to the left by 3 units.
- Thus, the vertical asymptote shifts from \( x = 0 \) to \( x = -3 \).
- The point \( (1, 0) \) of \( f(x) \) becomes the point \( (-2, 0) \) for \( g(x) \) because substituting \( -2 \) into \( g(x) \) gives \( g(-2) = \log_2(1) = 0 \).
Now graph \( g(x) \):
- Draw the vertical dashed line (asymptote) at \( x = -3 \).
- Plot the point \( (-2, 0) \).
- The graph will rise toward positive infinity as \( x \) goes toward infinity and approach negative infinity as \( x \) approaches \( -3 \) from the right.
### Part 2: Vertical Asymptote of \( g(x) \)
The vertical asymptote of \( g(x) \) is given by:
\[
x = -3
\]
### Part 3: Domain of \( g(x) \)
The function \( \log_2(x + 3) \) is defined when the argument \( x + 3 > 0 \):
\[
x + 3 > 0 \implies x > -3
\]
Thus, the domain of \( g(x) \) in interval notation is:
\[
(-3, \infty)
\]
### Part 4: Range of \( g(x) \)
The range of the logarithmic function \( g(x) = \log_2(x + 3) \) is all real numbers, as it can take any value from negative infinity to positive infinity, since the logarithm can grow indefinitely:
\[
(-\infty, \infty)
\]
### Summary of Responses
- **Vertical Asymptote**: \( x = -3 \)
- **Domain**: \( (-3, \infty) \)
- **Range**: \( (-\infty, \infty) \)
This completes the analysis of the function \( g(x) = \log_2(x + 3) \). You can now graph this function based on the information provided.