Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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.
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
