Answer :
Alright, let's break this down step-by-step for you:
### Part A: Type of Functions
Function [tex]\( f(x) = x^3 + x^2 - 3x + 4 \)[/tex]
To determine the type of the function [tex]\( f(x) \)[/tex], let's carefully examine its form. Here, [tex]\( f(x) \)[/tex] is expressed as a cubic polynomial where the highest power of [tex]\( x \)[/tex] is 3.
- Polynomial Function: A polynomial function is any function that can be written in the form [tex]\( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \)[/tex], where [tex]\( a_i \)[/tex] are constants.
- Since [tex]\( f(x) \)[/tex] has the form [tex]\( x^3 + x^2 - 3x + 4 \)[/tex], it clearly fits the criteria of a polynomial function.
- The highest degree of [tex]\( f(x) \)[/tex] is 3, making it a polynomial of degree 3.
So, [tex]\( f(x) \)[/tex] is a polynomial of degree 3.
Function [tex]\( g(x) = 2^x - 4 \)[/tex]
Now, let's look at [tex]\( g(x) \)[/tex].
- Exponential Function: An exponential function is any function of the form [tex]\( a^x \)[/tex] where [tex]\( a \)[/tex] is a positive constant other than 1 (in this case, 2).
- The function [tex]\( g(x) \)[/tex] involves the term [tex]\( 2^x \)[/tex], which is indeed an exponential expression.
- After the exponential term, we simply subtract 4, which does not change the nature of the function as being exponential.
Thus, [tex]\( g(x) \)[/tex] is an exponential function.
### Part B: Domain and Range
Domain and Range of [tex]\( f(x) = x^3 + x^2 - 3x + 4 \)[/tex]
- Domain: Polynomial functions are defined for all real numbers since you can input any real number for [tex]\( x \)[/tex] and get a valid output. Hence, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers}} \][/tex]
- Range: Polynomial functions of odd degree (like a cubic function) will extend from [tex]\(-\infty\)[/tex] to [tex]\(+\infty\)[/tex] because as [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex] or [tex]\( +\infty \)[/tex], the value of [tex]\( f(x) \)[/tex] also approaches [tex]\(-\infty \)[/tex] and [tex]\( +\infty \)[/tex] respectively. Thus, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers}} \][/tex]
Domain and Range of [tex]\( g(x) = 2^x - 4 \)[/tex]
- Domain: Exponential functions are defined for all real numbers as well. You can input any real number for [tex]\( x \)[/tex] and get a valid output. Hence, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers}} \][/tex]
- Range: For [tex]\( g(x) \)[/tex], since [tex]\( 2^x \)[/tex] is always positive (i.e., [tex]\( 2^x > 0 \)[/tex]), when you subtract 4 from it, the value will always be greater than [tex]\(-4\)[/tex] but never equal to [tex]\(-4\)[/tex]. Thus, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{(-4, \infty)} \][/tex]
### Comparison of Domain and Range
- Domain Comparison:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain, which is all real numbers.
- Range Comparison:
- The range of [tex]\( f(x) \)[/tex] is all real numbers.
- The range of [tex]\( g(x) \)[/tex] is [tex]\((-4, \infty)\)[/tex].
This detailed analysis shows that while both functions have the same domain, their ranges differ significantly. [tex]\( f(x) \)[/tex]'s values can take any real number whereas [tex]\( g(x) \)[/tex]'s values are constrained to always being greater than [tex]\(-4\)[/tex].
### Part A: Type of Functions
Function [tex]\( f(x) = x^3 + x^2 - 3x + 4 \)[/tex]
To determine the type of the function [tex]\( f(x) \)[/tex], let's carefully examine its form. Here, [tex]\( f(x) \)[/tex] is expressed as a cubic polynomial where the highest power of [tex]\( x \)[/tex] is 3.
- Polynomial Function: A polynomial function is any function that can be written in the form [tex]\( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \)[/tex], where [tex]\( a_i \)[/tex] are constants.
- Since [tex]\( f(x) \)[/tex] has the form [tex]\( x^3 + x^2 - 3x + 4 \)[/tex], it clearly fits the criteria of a polynomial function.
- The highest degree of [tex]\( f(x) \)[/tex] is 3, making it a polynomial of degree 3.
So, [tex]\( f(x) \)[/tex] is a polynomial of degree 3.
Function [tex]\( g(x) = 2^x - 4 \)[/tex]
Now, let's look at [tex]\( g(x) \)[/tex].
- Exponential Function: An exponential function is any function of the form [tex]\( a^x \)[/tex] where [tex]\( a \)[/tex] is a positive constant other than 1 (in this case, 2).
- The function [tex]\( g(x) \)[/tex] involves the term [tex]\( 2^x \)[/tex], which is indeed an exponential expression.
- After the exponential term, we simply subtract 4, which does not change the nature of the function as being exponential.
Thus, [tex]\( g(x) \)[/tex] is an exponential function.
### Part B: Domain and Range
Domain and Range of [tex]\( f(x) = x^3 + x^2 - 3x + 4 \)[/tex]
- Domain: Polynomial functions are defined for all real numbers since you can input any real number for [tex]\( x \)[/tex] and get a valid output. Hence, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers}} \][/tex]
- Range: Polynomial functions of odd degree (like a cubic function) will extend from [tex]\(-\infty\)[/tex] to [tex]\(+\infty\)[/tex] because as [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex] or [tex]\( +\infty \)[/tex], the value of [tex]\( f(x) \)[/tex] also approaches [tex]\(-\infty \)[/tex] and [tex]\( +\infty \)[/tex] respectively. Thus, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers}} \][/tex]
Domain and Range of [tex]\( g(x) = 2^x - 4 \)[/tex]
- Domain: Exponential functions are defined for all real numbers as well. You can input any real number for [tex]\( x \)[/tex] and get a valid output. Hence, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers}} \][/tex]
- Range: For [tex]\( g(x) \)[/tex], since [tex]\( 2^x \)[/tex] is always positive (i.e., [tex]\( 2^x > 0 \)[/tex]), when you subtract 4 from it, the value will always be greater than [tex]\(-4\)[/tex] but never equal to [tex]\(-4\)[/tex]. Thus, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{(-4, \infty)} \][/tex]
### Comparison of Domain and Range
- Domain Comparison:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain, which is all real numbers.
- Range Comparison:
- The range of [tex]\( f(x) \)[/tex] is all real numbers.
- The range of [tex]\( g(x) \)[/tex] is [tex]\((-4, \infty)\)[/tex].
This detailed analysis shows that while both functions have the same domain, their ranges differ significantly. [tex]\( f(x) \)[/tex]'s values can take any real number whereas [tex]\( g(x) \)[/tex]'s values are constrained to always being greater than [tex]\(-4\)[/tex].
