Answer :
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]
