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To thoroughly address the problem, we will analyze and describe both the nature and characteristics of the given functions [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex] and [tex]\( g(x) = \log(x) + 2 \)[/tex]. Our analysis will cover the type of each function, and their domain and range. Here's a step-by-step solution:
### Part A: Identify the types of functions
#### Function [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]:
1. Type of Function: This function is a polynomial.
- Justification: Polynomial functions are those of the form [tex]\( P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \)[/tex], where [tex]\( n \)[/tex] is a non-negative integer and all coefficients [tex]\( a_0, a_1, \ldots, a_n \)[/tex] are constants.
- For [tex]\( f(x) \)[/tex], the highest power of [tex]\( x \)[/tex] is 3, indicating it is a polynomial of degree 3.
#### Function [tex]\( g(x) = \log(x) + 2 \)[/tex]:
1. Type of Function: This function is a logarithmic function.
- Justification: Logarithmic functions are of the form [tex]\( g(x) = \log_b(x) \)[/tex], where [tex]\( b \)[/tex] is the base of the logarithm (commonly [tex]\( e \)[/tex] for natural logs or 10 for common logs). Since [tex]\( g(x) \)[/tex] includes [tex]\(\log(x)\)[/tex] plus a constant, it falls under the category of a logarithmic function.
### Part B: Determine the domain and range
#### Domain and Range for [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]:
1. Domain:
- Description: The domain of a polynomial function is all real numbers.
- Reason: Polynomials are defined for all real [tex]\( x \)[/tex] without any restrictions such as divisions by zero or taking logarithms of non-positive numbers.
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
2. Range:
- Description: The range of a polynomial function of odd degree (where the highest power term has an odd exponent) is all real numbers.
- Reason: Polynomials of odd degree go to [tex]\(\infty\)[/tex] as [tex]\( x \)[/tex] goes to [tex]\(\infty\)[/tex] and to [tex]\(-\infty\)[/tex] as [tex]\( x \)[/tex] goes to [tex]\(-\infty\)[/tex].
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
#### Domain and Range for [tex]\( g(x) = \log(x) + 2 \)[/tex]:
1. Domain:
- Description: The domain of logarithmic functions [tex]\( \log(x) \)[/tex] is [tex]\( x > 0 \)[/tex] (positive real numbers).
- Reason: Logarithms are only defined for positive values of [tex]\( x \)[/tex].
- Domain of [tex]\( g(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
2. Range:
- Description: The range of a logarithmic function [tex]\( \log(x) \)[/tex] is all real numbers.
- Reason: As [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( \log(x) \)[/tex] approaches [tex]\(-\infty\)[/tex], and as [tex]\( x \)[/tex] increases without bound, [tex]\( \log(x) \)[/tex] increases without bound.
- Range of [tex]\( g(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
### Comparison of Domains and Ranges
1. Domains:
- [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]: Domain is [tex]\( (-\infty, \infty) \)[/tex]
- [tex]\( g(x) = \log(x) + 2 \)[/tex]: Domain is [tex]\( (0, \infty) \)[/tex]
- Comparison: The domain of [tex]\( f(x) \)[/tex] is broader as it includes all real numbers. The domain of [tex]\( g(x) \)[/tex] is restricted to positive real numbers only.
2. Ranges:
- [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]: Range is [tex]\( (-\infty, \infty) \)[/tex]
- [tex]\( g(x) = \log(x) + 2 \)[/tex]: Range is [tex]\( (-\infty, \infty) \)[/tex]
- Comparison: Both functions share the same range, which is all real numbers ([tex]\( (-\infty, \infty) \)[/tex]).
Through this detailed analysis, we have identified the types, domains, and ranges of the given functions, along with a comparison of these mathematical properties.
### Part A: Identify the types of functions
#### Function [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]:
1. Type of Function: This function is a polynomial.
- Justification: Polynomial functions are those of the form [tex]\( P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \)[/tex], where [tex]\( n \)[/tex] is a non-negative integer and all coefficients [tex]\( a_0, a_1, \ldots, a_n \)[/tex] are constants.
- For [tex]\( f(x) \)[/tex], the highest power of [tex]\( x \)[/tex] is 3, indicating it is a polynomial of degree 3.
#### Function [tex]\( g(x) = \log(x) + 2 \)[/tex]:
1. Type of Function: This function is a logarithmic function.
- Justification: Logarithmic functions are of the form [tex]\( g(x) = \log_b(x) \)[/tex], where [tex]\( b \)[/tex] is the base of the logarithm (commonly [tex]\( e \)[/tex] for natural logs or 10 for common logs). Since [tex]\( g(x) \)[/tex] includes [tex]\(\log(x)\)[/tex] plus a constant, it falls under the category of a logarithmic function.
### Part B: Determine the domain and range
#### Domain and Range for [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]:
1. Domain:
- Description: The domain of a polynomial function is all real numbers.
- Reason: Polynomials are defined for all real [tex]\( x \)[/tex] without any restrictions such as divisions by zero or taking logarithms of non-positive numbers.
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
2. Range:
- Description: The range of a polynomial function of odd degree (where the highest power term has an odd exponent) is all real numbers.
- Reason: Polynomials of odd degree go to [tex]\(\infty\)[/tex] as [tex]\( x \)[/tex] goes to [tex]\(\infty\)[/tex] and to [tex]\(-\infty\)[/tex] as [tex]\( x \)[/tex] goes to [tex]\(-\infty\)[/tex].
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
#### Domain and Range for [tex]\( g(x) = \log(x) + 2 \)[/tex]:
1. Domain:
- Description: The domain of logarithmic functions [tex]\( \log(x) \)[/tex] is [tex]\( x > 0 \)[/tex] (positive real numbers).
- Reason: Logarithms are only defined for positive values of [tex]\( x \)[/tex].
- Domain of [tex]\( g(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
2. Range:
- Description: The range of a logarithmic function [tex]\( \log(x) \)[/tex] is all real numbers.
- Reason: As [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( \log(x) \)[/tex] approaches [tex]\(-\infty\)[/tex], and as [tex]\( x \)[/tex] increases without bound, [tex]\( \log(x) \)[/tex] increases without bound.
- Range of [tex]\( g(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
### Comparison of Domains and Ranges
1. Domains:
- [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]: Domain is [tex]\( (-\infty, \infty) \)[/tex]
- [tex]\( g(x) = \log(x) + 2 \)[/tex]: Domain is [tex]\( (0, \infty) \)[/tex]
- Comparison: The domain of [tex]\( f(x) \)[/tex] is broader as it includes all real numbers. The domain of [tex]\( g(x) \)[/tex] is restricted to positive real numbers only.
2. Ranges:
- [tex]\( f(x) = x^3 + x^2 - 2x + 3 \)[/tex]: Range is [tex]\( (-\infty, \infty) \)[/tex]
- [tex]\( g(x) = \log(x) + 2 \)[/tex]: Range is [tex]\( (-\infty, \infty) \)[/tex]
- Comparison: Both functions share the same range, which is all real numbers ([tex]\( (-\infty, \infty) \)[/tex]).
Through this detailed analysis, we have identified the types, domains, and ranges of the given functions, along with a comparison of these mathematical properties.
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