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To graph the logarithmic function [tex]\( g(x) = -2 + \log_4(x + 1) \)[/tex], follow these steps:
1. Identify the basic form and transformations:
- The base function here is [tex]\( \log_4(x) \)[/tex].
- The given function [tex]\( g(x) = -2 + \log_4(x + 1) \)[/tex] involves two transformations:
- A horizontal shift to the left by 1 unit ([tex]\( x \rightarrow x + 1 \)[/tex]).
- A vertical shift downward by 2 units.
2. Identify the vertical asymptote:
- For the function [tex]\( \log_4(x+1) \)[/tex]:
- The logarithm is undefined when its argument is less than or equal to 0.
- Therefore, [tex]\( x + 1 = 0 \implies x = -1 \)[/tex].
- The vertical asymptote is [tex]\( x = -1 \)[/tex].
3. Plot key points:
- Find two points on the graph for easier plotting.
- Choose values for [tex]\( x \)[/tex] and determine [tex]\( g(x) \)[/tex].
a) Find [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = -2 + \log_4(0 + 1) = -2 + \log_4(1) = -2 + 0 = -2 \][/tex]
Point: [tex]\( (0, -2) \)[/tex]
b) Find [tex]\( g(3) \)[/tex]:
[tex]\[ g(3) = -2 + \log_4(3 + 1) = -2 + \log_4(4) = -2 + 1 = -1 \][/tex]
Point: [tex]\( (3, -1) \)[/tex]
4. Draw the vertical asymptote:
- Draw a dashed line at [tex]\( x = -1 \)[/tex].
5. Plot the points:
- Plot the points [tex]\( (0, -2) \)[/tex] and [tex]\( (3, -1) \)[/tex].
6. Draw the curve:
- Sketch a smooth curve through the points, approaching the vertical asymptote [tex]\( x = -1 \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -1 \)[/tex] from the right.
7. Indicate the domain and range:
- Domain: The function is defined for [tex]\( x > -1 \)[/tex], so the domain in interval notation is:
[tex]\[ (-1, \infty) \][/tex]
- Range: Since a logarithmic function can take any real value and our function is shifted downward by 2 units, the range is all real numbers. The range in interval notation is:
[tex]\[ (-\infty, \infty) \][/tex]
Now, you have the graph and the requested points and vertical asymptote. Ensure to use accurate graphing tools to plot this function and properly label the asymptote, points, and axes.
Domain: [tex]\( (-1, \infty) \)[/tex]
Range: [tex]\( (-\infty, \infty) \)[/tex]
1. Identify the basic form and transformations:
- The base function here is [tex]\( \log_4(x) \)[/tex].
- The given function [tex]\( g(x) = -2 + \log_4(x + 1) \)[/tex] involves two transformations:
- A horizontal shift to the left by 1 unit ([tex]\( x \rightarrow x + 1 \)[/tex]).
- A vertical shift downward by 2 units.
2. Identify the vertical asymptote:
- For the function [tex]\( \log_4(x+1) \)[/tex]:
- The logarithm is undefined when its argument is less than or equal to 0.
- Therefore, [tex]\( x + 1 = 0 \implies x = -1 \)[/tex].
- The vertical asymptote is [tex]\( x = -1 \)[/tex].
3. Plot key points:
- Find two points on the graph for easier plotting.
- Choose values for [tex]\( x \)[/tex] and determine [tex]\( g(x) \)[/tex].
a) Find [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = -2 + \log_4(0 + 1) = -2 + \log_4(1) = -2 + 0 = -2 \][/tex]
Point: [tex]\( (0, -2) \)[/tex]
b) Find [tex]\( g(3) \)[/tex]:
[tex]\[ g(3) = -2 + \log_4(3 + 1) = -2 + \log_4(4) = -2 + 1 = -1 \][/tex]
Point: [tex]\( (3, -1) \)[/tex]
4. Draw the vertical asymptote:
- Draw a dashed line at [tex]\( x = -1 \)[/tex].
5. Plot the points:
- Plot the points [tex]\( (0, -2) \)[/tex] and [tex]\( (3, -1) \)[/tex].
6. Draw the curve:
- Sketch a smooth curve through the points, approaching the vertical asymptote [tex]\( x = -1 \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -1 \)[/tex] from the right.
7. Indicate the domain and range:
- Domain: The function is defined for [tex]\( x > -1 \)[/tex], so the domain in interval notation is:
[tex]\[ (-1, \infty) \][/tex]
- Range: Since a logarithmic function can take any real value and our function is shifted downward by 2 units, the range is all real numbers. The range in interval notation is:
[tex]\[ (-\infty, \infty) \][/tex]
Now, you have the graph and the requested points and vertical asymptote. Ensure to use accurate graphing tools to plot this function and properly label the asymptote, points, and axes.
Domain: [tex]\( (-1, \infty) \)[/tex]
Range: [tex]\( (-\infty, \infty) \)[/tex]
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