Answer :
To solve this problem, we'll analyze the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] step by step.
Part A: Identify the types of functions
1. Function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^3 + x^2 - 3x + 4 \][/tex]
To determine the type of this function, we observe the terms.
- [tex]\( x^3 \)[/tex] is a cubic term.
- [tex]\( x^2 \)[/tex] is a quadratic term.
- [tex]\( -3x \)[/tex] is a linear term.
- [tex]\( +4 \)[/tex] is a constant term.
The highest power of [tex]\( x \)[/tex] is 3, which makes it a polynomial function of degree 3. Therefore, [tex]\( f(x) \)[/tex] is a polynomial function.
2. Function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 2^x - 4 \][/tex]
Here, we have a term [tex]\( 2^x \)[/tex], which is an exponential term with base 2. The function is of the form [tex]\( a^x \)[/tex] where [tex]\( a > 0 \)[/tex] (in this case [tex]\( a = 2 \)[/tex]), minus a constant 4. Therefore, [tex]\( g(x) \)[/tex] is an exponential function.
Part B: Determine the domain and range
1. Function [tex]\( f(x) \)[/tex]:
- Domain:
A polynomial function can take any real number as input. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
[tex]\[ \text{Domain of } f(x): \, (-\infty, \infty) \, \text{or simply} \, \mathbb{R} \][/tex]
- Range:
For a polynomial of degree 3 (cubic polynomial), as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], the value of the function also approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], respectively. Hence, a cubic polynomial can take any real number as its range.
[tex]\[ \text{Range of } f(x): \, (-\infty, \infty) \, \text{or simply} \, \mathbb{R} \][/tex]
2. Function [tex]\( g(x) \)[/tex]:
- Domain:
An exponential function with the form [tex]\( 2^x - 4 \)[/tex] can take any real number as input. Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers.
[tex]\[ \text{Domain of } g(x): \, (-\infty, \infty) \, \text{or simply} \, \mathbb{R} \][/tex]
- Range:
The base of the exponential function [tex]\( 2^x \)[/tex] is always positive. The smallest value [tex]\( 2^x \)[/tex] can approach is 0 (as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]), but it never actually reaches 0. Subtracting 4 from [tex]\( 2^x \)[/tex], the smallest possible value is when [tex]\( x \)[/tex] is very large and negative, yielding [tex]\( 2^x \approx 0 \)[/tex], giving [tex]\( 2^x - 4 \approx -4 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] increases with no upper bound. Therefore, the range of [tex]\( g(x) \)[/tex] starts from [tex]\(-4\)[/tex] and increases to [tex]\( \infty \)[/tex].
[tex]\[ \text{Range of } g(x): \, (-4, \infty) \, \text{or} \, (-4, \infty) \][/tex]
Comparison:
- Domain Comparison:
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain, as both can take any real number as input.
[tex]\[ \text{Domain comparison: } \mathbb{R} \text{ for both functions} \][/tex]
- Range Comparison:
Unlike their domains, the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are different.
- The range of [tex]\( f(x) \)[/tex] is all real numbers: [tex]\(\mathbb{R}\)[/tex].
- The range of [tex]\( g(x) \)[/tex] is from [tex]\(-4\)[/tex] to [tex]\(\infty\)[/tex]:
[tex]\[ \text{Range of } f(x): \, \mathbb{R} \][/tex]
[tex]\[ \text{Range of } g(x): \, [-4, \infty) \][/tex]
Therefore, the range of [tex]\( g(x) \)[/tex] is a subset of the range of [tex]\( f(x) \)[/tex].
In summary, [tex]\( f(x) \)[/tex] is a polynomial function with domain and range both equal to all real numbers, while [tex]\( g(x) \)[/tex] is an exponential function with domain of all real numbers and range from [tex]\(-4\)[/tex] to [tex]\(\infty\)[/tex].
Part A: Identify the types of functions
1. Function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^3 + x^2 - 3x + 4 \][/tex]
To determine the type of this function, we observe the terms.
- [tex]\( x^3 \)[/tex] is a cubic term.
- [tex]\( x^2 \)[/tex] is a quadratic term.
- [tex]\( -3x \)[/tex] is a linear term.
- [tex]\( +4 \)[/tex] is a constant term.
The highest power of [tex]\( x \)[/tex] is 3, which makes it a polynomial function of degree 3. Therefore, [tex]\( f(x) \)[/tex] is a polynomial function.
2. Function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 2^x - 4 \][/tex]
Here, we have a term [tex]\( 2^x \)[/tex], which is an exponential term with base 2. The function is of the form [tex]\( a^x \)[/tex] where [tex]\( a > 0 \)[/tex] (in this case [tex]\( a = 2 \)[/tex]), minus a constant 4. Therefore, [tex]\( g(x) \)[/tex] is an exponential function.
Part B: Determine the domain and range
1. Function [tex]\( f(x) \)[/tex]:
- Domain:
A polynomial function can take any real number as input. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
[tex]\[ \text{Domain of } f(x): \, (-\infty, \infty) \, \text{or simply} \, \mathbb{R} \][/tex]
- Range:
For a polynomial of degree 3 (cubic polynomial), as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], the value of the function also approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], respectively. Hence, a cubic polynomial can take any real number as its range.
[tex]\[ \text{Range of } f(x): \, (-\infty, \infty) \, \text{or simply} \, \mathbb{R} \][/tex]
2. Function [tex]\( g(x) \)[/tex]:
- Domain:
An exponential function with the form [tex]\( 2^x - 4 \)[/tex] can take any real number as input. Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers.
[tex]\[ \text{Domain of } g(x): \, (-\infty, \infty) \, \text{or simply} \, \mathbb{R} \][/tex]
- Range:
The base of the exponential function [tex]\( 2^x \)[/tex] is always positive. The smallest value [tex]\( 2^x \)[/tex] can approach is 0 (as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]), but it never actually reaches 0. Subtracting 4 from [tex]\( 2^x \)[/tex], the smallest possible value is when [tex]\( x \)[/tex] is very large and negative, yielding [tex]\( 2^x \approx 0 \)[/tex], giving [tex]\( 2^x - 4 \approx -4 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] increases with no upper bound. Therefore, the range of [tex]\( g(x) \)[/tex] starts from [tex]\(-4\)[/tex] and increases to [tex]\( \infty \)[/tex].
[tex]\[ \text{Range of } g(x): \, (-4, \infty) \, \text{or} \, (-4, \infty) \][/tex]
Comparison:
- Domain Comparison:
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain, as both can take any real number as input.
[tex]\[ \text{Domain comparison: } \mathbb{R} \text{ for both functions} \][/tex]
- Range Comparison:
Unlike their domains, the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are different.
- The range of [tex]\( f(x) \)[/tex] is all real numbers: [tex]\(\mathbb{R}\)[/tex].
- The range of [tex]\( g(x) \)[/tex] is from [tex]\(-4\)[/tex] to [tex]\(\infty\)[/tex]:
[tex]\[ \text{Range of } f(x): \, \mathbb{R} \][/tex]
[tex]\[ \text{Range of } g(x): \, [-4, \infty) \][/tex]
Therefore, the range of [tex]\( g(x) \)[/tex] is a subset of the range of [tex]\( f(x) \)[/tex].
In summary, [tex]\( f(x) \)[/tex] is a polynomial function with domain and range both equal to all real numbers, while [tex]\( g(x) \)[/tex] is an exponential function with domain of all real numbers and range from [tex]\(-4\)[/tex] to [tex]\(\infty\)[/tex].
