Math Problem Statement
Solution
The problem requires you to identify the domain, range, and asymptote of the given logarithmic functions, as well as sketch their graphs and describe the transformations applied.
Let's go through each function individually:
17)
Base function: The parent function is .
Transformations:
- The inside the logarithm shifts the graph 3 units to the left.
- The outside the logarithm shifts the graph 4 units upwards.
Domain: For the logarithmic function to be defined, the argument , so the domain is: In interval notation, the domain is:
Range: The range of any logarithmic function is always all real numbers, so:
Vertical Asymptote: The vertical asymptote is where the argument inside the logarithm equals zero, so:
Sketch: You should draw the graph of , showing the vertical asymptote at , the shift left by 3 units, and up by 4 units. The graph will approach the vertical asymptote but never touch it.
18)
Base function: The parent function is .
Transformations:
- The inside the logarithm shifts the graph 1 unit to the right.
- The outside the logarithm shifts the graph 4 units downwards.
Domain: For the logarithmic function to be defined, the argument , so the domain is: In interval notation, the domain is:
Range: As with all logarithmic functions, the range is all real numbers:
Vertical Asymptote: The vertical asymptote is where the argument inside the logarithm equals zero, so:
Sketch: You should draw the graph of , showing the vertical asymptote at , the shift right by 1 unit, and down by 4 units. Again, the graph will approach the vertical asymptote but will not touch it.
Would you like further clarification or more details on these problems?
Here are 5 related questions:
- How do different bases of logarithmic functions affect the steepness of the graph?
- What happens if we reflect a logarithmic function across the x-axis or y-axis?
- How do we determine the horizontal asymptote for other types of functions?
- What if we apply a negative transformation inside the logarithm, like ?
- How do vertical and horizontal stretches impact logarithmic graphs?
Tip: Always check the domain of logarithmic functions by solving so that the argument inside the logarithm is positive.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Functions
Transformations
Domain and Range
Vertical Asymptote
Formulas
y = log_b(x)
Vertical asymptote: x = a when the function is y = log_b(x - a)
Theorems
Properties of Logarithms
Suitable Grade Level
Grades 10-12
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