Math Problem Statement
Реши задачу с картинки и построй единый график
Solution
The problem in the image asks for a complete investigation and plotting of the following three functions:
To solve this problem, we need to analyze each function separately and then plot them on a single graph. Let's start by examining each function.
1. Function Analysis:
- Domain: The function is undefined when the denominator is zero, i.e., . Solving this, we get . So, the domain is .
- Asymptotes:
- Vertical asymptotes: At .
- Horizontal asymptote: As , , indicating no horizontal asymptote. Instead, there is an oblique asymptote.
- Oblique asymptote: Divide by . Using polynomial division: the oblique asymptote is .
- Derivative (Critical points): .
- Critical points: and .
2. Function Analysis:
- Domain: The function is defined for all , so the domain is .
- Asymptotes:
- Vertical asymptote: At .
- Horizontal asymptote: None. As , , and as , as well.
- Derivative (Critical points): .
- For , (decreasing function).
- For , (increasing function).
3. Function Analysis:
- Domain: The function is defined for all , .
- Range: Since is always positive, . The range of depends on the range of , which is . Hence, varies from to .
- Periodicity: This function is periodic with period , same as .
- Derivative (Critical points): .
- when , i.e., (where is an integer).
Graphing All Functions Together
- Plot each function, marking the critical points, asymptotes, and behavior at infinity.
- Ensure the graph captures the periodic nature, increasing and decreasing intervals, and vertical asymptotes where applicable.
Would you like more details on a specific part of the analysis or see the graphs for these functions?
Here are some related questions you might find useful:
- What are the steps to find the oblique asymptote for a rational function?
- How does the derivative determine the increasing and decreasing behavior of a function?
- What methods are used to find critical points of a function?
- How do periodic functions behave differently from non-periodic functions?
- What are the key characteristics to look for when sketching graphs of exponential functions?
Tip: For periodic functions, always check for the range of values over one complete period to understand the repeating behavior.
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Math Problem Analysis
Mathematical Concepts
Functions
Exponential Functions
Trigonometric Functions
Graphing Functions
Asymptotes
Derivatives
Formulas
Exponential function formula: y = e^x
Derivative formulas for rational and exponential functions
Theorems
Properties of exponential functions
Behavior of trigonometric functions
Suitable Grade Level
Advanced High School
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