Answer :
### Part A: Identifying the Type of Functions
Function [tex]\( f(x) = -4^x + 5 \)[/tex]:
- The function [tex]\( f(x) \)[/tex] takes on the form [tex]\( a^x + c \)[/tex] where [tex]\( a = -4 \)[/tex] and [tex]\( c = 5 \)[/tex].
- This is characteristic of an exponential function, where an exponential function is defined generically as [tex]\( f(x) = a^x \)[/tex] where [tex]\( a \)[/tex] is a constant base.
- Specifically, [tex]\( f(x) \)[/tex] involves an exponential term raised to the power of [tex]\( x \)[/tex], modified by a constant shifting of +5.
Thus, [tex]\( f(x) = -4^x + 5 \)[/tex] is an exponential function.
Function [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- The function [tex]\( g(x) \)[/tex] consists of terms that are powers of [tex]\( x \)[/tex] with non-negative integer exponents [tex]\( (3, 2, 1, 0) \)[/tex].
- This is characteristic of a polynomial function, which can be written generally as [tex]\( g(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \)[/tex], where [tex]\( n \)[/tex] is a non-negative integer.
Thus, [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex] is a polynomial function.
### Part B: Finding the Domain and Range of the Functions
Domain of [tex]\( f(x) \)[/tex]:
- Exponential functions are defined for all real numbers. There are no restrictions on the values that [tex]\( x \)[/tex] can take for [tex]\( f(x) = -4^x + 5 \)[/tex].
Thus, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
Range of [tex]\( f(x) \)[/tex]:
- Considering [tex]\( f(x) = -4^x + 5 \)[/tex]:
- As [tex]\( x \to \infty \)[/tex], [tex]\( 4^x \to \infty \)[/tex] and hence [tex]\( -4^x \to -\infty \)[/tex]. Adding 5 still drives the function towards negative infinity.
- As [tex]\( x \to -\infty \)[/tex], [tex]\( 4^x \to 0 \)[/tex] and so [tex]\( -4^x \to 0 \)[/tex]; thus, [tex]\( f(x) \to 5 \)[/tex].
Since the function's value decreases without bound as [tex]\( x \)[/tex] approaches positive infinity and approaches 5 from below as [tex]\( x \)[/tex] approaches negative infinity, the range is [tex]\( (-\infty, 5) \)[/tex].
Domain of [tex]\( g(x) \)[/tex]:
- Polynomial functions are defined for all real numbers.
Thus, the domain of [tex]\( g(x) \)[/tex] is all real numbers.
Range of [tex]\( g(x) \)[/tex]:
- A polynomial of degree 3 (cubic polynomial) spans all real numbers, meaning for sufficiently large positive and negative values of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] can take any real value.
Thus, the range of [tex]\( g(x) \)[/tex] is all real numbers.
### Comparison of Domains and Ranges
Comparison of Domains:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers.
- The domain of [tex]\( g(x) \)[/tex] is all real numbers.
Since both domains are all real numbers, the domains are the same.
Comparison of Ranges:
- The range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 5) \)[/tex].
- The range of [tex]\( g(x) \)[/tex] is all real numbers.
Since these ranges are not the same, the ranges differ.
### Summary
- Type of [tex]\( f(x) \)[/tex]: Exponential
- Type of [tex]\( g(x) \)[/tex]: Polynomial
- Domain of [tex]\( f(x) \)[/tex]: All real numbers
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, 5) \)[/tex]
- Domain of [tex]\( g(x) \)[/tex]: All real numbers
- Range of [tex]\( g(x) \)[/tex]: All real numbers
- Comparison of Domains: Same
- Comparison of Ranges: Different
Function [tex]\( f(x) = -4^x + 5 \)[/tex]:
- The function [tex]\( f(x) \)[/tex] takes on the form [tex]\( a^x + c \)[/tex] where [tex]\( a = -4 \)[/tex] and [tex]\( c = 5 \)[/tex].
- This is characteristic of an exponential function, where an exponential function is defined generically as [tex]\( f(x) = a^x \)[/tex] where [tex]\( a \)[/tex] is a constant base.
- Specifically, [tex]\( f(x) \)[/tex] involves an exponential term raised to the power of [tex]\( x \)[/tex], modified by a constant shifting of +5.
Thus, [tex]\( f(x) = -4^x + 5 \)[/tex] is an exponential function.
Function [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- The function [tex]\( g(x) \)[/tex] consists of terms that are powers of [tex]\( x \)[/tex] with non-negative integer exponents [tex]\( (3, 2, 1, 0) \)[/tex].
- This is characteristic of a polynomial function, which can be written generally as [tex]\( g(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \)[/tex], where [tex]\( n \)[/tex] is a non-negative integer.
Thus, [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex] is a polynomial function.
### Part B: Finding the Domain and Range of the Functions
Domain of [tex]\( f(x) \)[/tex]:
- Exponential functions are defined for all real numbers. There are no restrictions on the values that [tex]\( x \)[/tex] can take for [tex]\( f(x) = -4^x + 5 \)[/tex].
Thus, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
Range of [tex]\( f(x) \)[/tex]:
- Considering [tex]\( f(x) = -4^x + 5 \)[/tex]:
- As [tex]\( x \to \infty \)[/tex], [tex]\( 4^x \to \infty \)[/tex] and hence [tex]\( -4^x \to -\infty \)[/tex]. Adding 5 still drives the function towards negative infinity.
- As [tex]\( x \to -\infty \)[/tex], [tex]\( 4^x \to 0 \)[/tex] and so [tex]\( -4^x \to 0 \)[/tex]; thus, [tex]\( f(x) \to 5 \)[/tex].
Since the function's value decreases without bound as [tex]\( x \)[/tex] approaches positive infinity and approaches 5 from below as [tex]\( x \)[/tex] approaches negative infinity, the range is [tex]\( (-\infty, 5) \)[/tex].
Domain of [tex]\( g(x) \)[/tex]:
- Polynomial functions are defined for all real numbers.
Thus, the domain of [tex]\( g(x) \)[/tex] is all real numbers.
Range of [tex]\( g(x) \)[/tex]:
- A polynomial of degree 3 (cubic polynomial) spans all real numbers, meaning for sufficiently large positive and negative values of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] can take any real value.
Thus, the range of [tex]\( g(x) \)[/tex] is all real numbers.
### Comparison of Domains and Ranges
Comparison of Domains:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers.
- The domain of [tex]\( g(x) \)[/tex] is all real numbers.
Since both domains are all real numbers, the domains are the same.
Comparison of Ranges:
- The range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 5) \)[/tex].
- The range of [tex]\( g(x) \)[/tex] is all real numbers.
Since these ranges are not the same, the ranges differ.
### Summary
- Type of [tex]\( f(x) \)[/tex]: Exponential
- Type of [tex]\( g(x) \)[/tex]: Polynomial
- Domain of [tex]\( f(x) \)[/tex]: All real numbers
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, 5) \)[/tex]
- Domain of [tex]\( g(x) \)[/tex]: All real numbers
- Range of [tex]\( g(x) \)[/tex]: All real numbers
- Comparison of Domains: Same
- Comparison of Ranges: Different
