Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Sure! Let's break down the problem into detailed steps and address each part accordingly.
### Part A: Identifying the Types of Functions
Function [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]:
- Explanation:
- The function [tex]\( f(x) \)[/tex] is given as [tex]\( x^3 + x^2 - 2x + 3 \)[/tex].
- This expression is a sum of terms involving powers of [tex]\( x \)[/tex] with coefficients: [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], [tex]\(-2x\)[/tex], and the constant term [tex]\( +3 \)[/tex].
- Each term involves [tex]\( x \)[/tex] raised to a non-negative integer power.
- Conclusion:
- Since [tex]\( f(x) \)[/tex] is expressed as a polynomial of degree 3, it is a polynomial function.
Function [tex]\( g(x) = \log(x) + 2 \)[/tex]:
- Explanation:
- The function [tex]\( g(x) \)[/tex] is given as [tex]\( \log(x) + 2 \)[/tex].
- This consists of the natural logarithm function [tex]\( \log(x) \)[/tex] plus a constant [tex]\( 2 \)[/tex].
- The natural logarithm function, [tex]\( \log(x) \)[/tex], is defined for [tex]\( x > 0 \)[/tex].
- Conclusion:
- Since [tex]\( g(x) \)[/tex] involves the logarithmic function as its primary component, it is a logarithmic function.
### Part B: Finding the Domain and Range
Domain and Range of [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]
- Domain:
- For polynomial functions, there are no restrictions on the value of [tex]\( x \)[/tex].
- Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Range:
- Polynomial functions, particularly those of odd degree (like the cubic polynomial in this case), cover all possible real values as [tex]\( x \)[/tex] ranges over all real numbers.
- Therefore, the range of [tex]\( f(x) \)[/tex] is all real numbers.
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
Domain and Range of [tex]\( g(x) = \log(x) + 2 \)[/tex]
- Domain:
- The logarithmic function [tex]\( \log(x) \)[/tex] is defined only for [tex]\( x > 0 \)[/tex].
- Therefore, the domain of [tex]\( g(x) \)[/tex] is all positive real numbers.
- Domain of [tex]\( g(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Range:
- The logarithmic function [tex]\( \log(x) \)[/tex] can take any real value [tex]\( (-\infty, \infty) \)[/tex], and adding 2 to it does not change this fact.
- Therefore, the range of [tex]\( g(x) \)[/tex] is all real numbers.
- Range of [tex]\( g(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
### Comparing Domains and Ranges
- Comparing Domains:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex], meaning it includes all real numbers.
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex], meaning it includes all positive real numbers.
- Comparison: [tex]\( f(x) \)[/tex] has domain [tex]\( (-\infty, \infty) \)[/tex], while [tex]\( g(x) \)[/tex] has domain [tex]\( (0, \infty) \)[/tex].
- Comparing Ranges:
- The range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex], covering all real numbers.
- The range of [tex]\( g(x) \)[/tex] is also [tex]\( (-\infty, \infty) \)[/tex], covering all real numbers.
- Comparison: Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have range [tex]\( (-\infty, \infty) \)[/tex].
To summarize:
- Types of Functions:
- [tex]\( f(x) \)[/tex] is a polynomial function.
- [tex]\( g(x) \)[/tex] is a logarithmic function.
- Domains:
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Domain of [tex]\( g(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Ranges:
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Range of [tex]\( g(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Comparisons:
- Domains: [tex]\( f(x) \)[/tex] has domain [tex]\( (-\infty, \infty) \)[/tex] while [tex]\( g(x) \)[/tex] has domain [tex]\( (0, \infty) \)[/tex]
- Ranges: Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have range [tex]\( (-\infty, \infty) \)[/tex]
### Part A: Identifying the Types of Functions
Function [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]:
- Explanation:
- The function [tex]\( f(x) \)[/tex] is given as [tex]\( x^3 + x^2 - 2x + 3 \)[/tex].
- This expression is a sum of terms involving powers of [tex]\( x \)[/tex] with coefficients: [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], [tex]\(-2x\)[/tex], and the constant term [tex]\( +3 \)[/tex].
- Each term involves [tex]\( x \)[/tex] raised to a non-negative integer power.
- Conclusion:
- Since [tex]\( f(x) \)[/tex] is expressed as a polynomial of degree 3, it is a polynomial function.
Function [tex]\( g(x) = \log(x) + 2 \)[/tex]:
- Explanation:
- The function [tex]\( g(x) \)[/tex] is given as [tex]\( \log(x) + 2 \)[/tex].
- This consists of the natural logarithm function [tex]\( \log(x) \)[/tex] plus a constant [tex]\( 2 \)[/tex].
- The natural logarithm function, [tex]\( \log(x) \)[/tex], is defined for [tex]\( x > 0 \)[/tex].
- Conclusion:
- Since [tex]\( g(x) \)[/tex] involves the logarithmic function as its primary component, it is a logarithmic function.
### Part B: Finding the Domain and Range
Domain and Range of [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]
- Domain:
- For polynomial functions, there are no restrictions on the value of [tex]\( x \)[/tex].
- Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Range:
- Polynomial functions, particularly those of odd degree (like the cubic polynomial in this case), cover all possible real values as [tex]\( x \)[/tex] ranges over all real numbers.
- Therefore, the range of [tex]\( f(x) \)[/tex] is all real numbers.
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
Domain and Range of [tex]\( g(x) = \log(x) + 2 \)[/tex]
- Domain:
- The logarithmic function [tex]\( \log(x) \)[/tex] is defined only for [tex]\( x > 0 \)[/tex].
- Therefore, the domain of [tex]\( g(x) \)[/tex] is all positive real numbers.
- Domain of [tex]\( g(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Range:
- The logarithmic function [tex]\( \log(x) \)[/tex] can take any real value [tex]\( (-\infty, \infty) \)[/tex], and adding 2 to it does not change this fact.
- Therefore, the range of [tex]\( g(x) \)[/tex] is all real numbers.
- Range of [tex]\( g(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
### Comparing Domains and Ranges
- Comparing Domains:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex], meaning it includes all real numbers.
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex], meaning it includes all positive real numbers.
- Comparison: [tex]\( f(x) \)[/tex] has domain [tex]\( (-\infty, \infty) \)[/tex], while [tex]\( g(x) \)[/tex] has domain [tex]\( (0, \infty) \)[/tex].
- Comparing Ranges:
- The range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex], covering all real numbers.
- The range of [tex]\( g(x) \)[/tex] is also [tex]\( (-\infty, \infty) \)[/tex], covering all real numbers.
- Comparison: Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have range [tex]\( (-\infty, \infty) \)[/tex].
To summarize:
- Types of Functions:
- [tex]\( f(x) \)[/tex] is a polynomial function.
- [tex]\( g(x) \)[/tex] is a logarithmic function.
- Domains:
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Domain of [tex]\( g(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Ranges:
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Range of [tex]\( g(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Comparisons:
- Domains: [tex]\( f(x) \)[/tex] has domain [tex]\( (-\infty, \infty) \)[/tex] while [tex]\( g(x) \)[/tex] has domain [tex]\( (0, \infty) \)[/tex]
- Ranges: Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have range [tex]\( (-\infty, \infty) \)[/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
