Answer :
Let's address each part of the question in detail:
### Part A: Type of Functions
- For [tex]\( f(x) = -4^x + 5 \)[/tex]:
- This is an exponential function. The defining characteristic of an exponential function is that the variable [tex]\( x \)[/tex] is in the exponent. In this case, [tex]\( -4^x \)[/tex] denotes an exponential term.
- For [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- This is a polynomial function. A polynomial is defined as a sum of terms, each of which consists of a variable raised to a non-negative integer power multiplied by a constant. Here, [tex]\( g(x) \)[/tex] is composed of terms [tex]\( x^3, x^2, -4x \)[/tex], and a constant 5, which makes it a polynomial of degree 3.
### Part B: Domain and Range
- Domain:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- For [tex]\( f(x) = -4^x + 5 \)[/tex]:
- The domain of an exponential function [tex]\( -4^x + 5 \)[/tex] is all real numbers, [tex]\( x \in (-\infty, \infty) \)[/tex], because you can exponentiate any real number.
- For [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- The domain of a polynomial function is also all real numbers, [tex]\( x \in (-\infty, \infty) \)[/tex], since polynomials are defined for all real numbers.
- Range:
- The range of a function is the set of all possible output values (y-values) that the function can produce.
- For [tex]\( f(x) = -4^x + 5 \)[/tex]:
- To find the range, note that the exponential function [tex]\( 4^x \)[/tex] grows very rapidly as [tex]\( x \)[/tex] increases and approaches zero as [tex]\( x \)[/tex] decreases. Consequently, [tex]\( -4^x \)[/tex] will very rapidly decrease (since the exponent is negative). Therefore:
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( -4^x \)[/tex] approaches [tex]\( -\infty \)[/tex], so [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( -4^x \)[/tex] approaches 0, so [tex]\( f(x) \)[/tex] approaches 5 from below.
- Thus, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 5) \)[/tex].
- For [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- Since this is a cubic polynomial, it is known that the range of any cubic polynomial is all real numbers (because the leading term [tex]\( x^3 \)[/tex] dominates as [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex]). So, the range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
### Comparison of Domains and Ranges
- Domains:
- As noted, both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain: [tex]\( (-\infty, \infty) \)[/tex].
- Ranges:
- The range of [tex]\( f(x) \)[/tex] (exponential function) is [tex]\( (-\infty, 5) \)[/tex], which is all values less than 5.
- The range of [tex]\( g(x) \)[/tex] (polynomial function) is [tex]\( (-\infty, \infty) \)[/tex], which includes all real numbers.
In summary, while both functions share the same domain of all real numbers, their ranges differ: [tex]\( f(x) \)[/tex] is restricted to values less than 5, while [tex]\( g(x) \)[/tex] includes all real numbers.
### Part A: Type of Functions
- For [tex]\( f(x) = -4^x + 5 \)[/tex]:
- This is an exponential function. The defining characteristic of an exponential function is that the variable [tex]\( x \)[/tex] is in the exponent. In this case, [tex]\( -4^x \)[/tex] denotes an exponential term.
- For [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- This is a polynomial function. A polynomial is defined as a sum of terms, each of which consists of a variable raised to a non-negative integer power multiplied by a constant. Here, [tex]\( g(x) \)[/tex] is composed of terms [tex]\( x^3, x^2, -4x \)[/tex], and a constant 5, which makes it a polynomial of degree 3.
### Part B: Domain and Range
- Domain:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- For [tex]\( f(x) = -4^x + 5 \)[/tex]:
- The domain of an exponential function [tex]\( -4^x + 5 \)[/tex] is all real numbers, [tex]\( x \in (-\infty, \infty) \)[/tex], because you can exponentiate any real number.
- For [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- The domain of a polynomial function is also all real numbers, [tex]\( x \in (-\infty, \infty) \)[/tex], since polynomials are defined for all real numbers.
- Range:
- The range of a function is the set of all possible output values (y-values) that the function can produce.
- For [tex]\( f(x) = -4^x + 5 \)[/tex]:
- To find the range, note that the exponential function [tex]\( 4^x \)[/tex] grows very rapidly as [tex]\( x \)[/tex] increases and approaches zero as [tex]\( x \)[/tex] decreases. Consequently, [tex]\( -4^x \)[/tex] will very rapidly decrease (since the exponent is negative). Therefore:
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( -4^x \)[/tex] approaches [tex]\( -\infty \)[/tex], so [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( -4^x \)[/tex] approaches 0, so [tex]\( f(x) \)[/tex] approaches 5 from below.
- Thus, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 5) \)[/tex].
- For [tex]\( g(x) = x^3 + x^2 - 4x + 5 \)[/tex]:
- Since this is a cubic polynomial, it is known that the range of any cubic polynomial is all real numbers (because the leading term [tex]\( x^3 \)[/tex] dominates as [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex]). So, the range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
### Comparison of Domains and Ranges
- Domains:
- As noted, both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain: [tex]\( (-\infty, \infty) \)[/tex].
- Ranges:
- The range of [tex]\( f(x) \)[/tex] (exponential function) is [tex]\( (-\infty, 5) \)[/tex], which is all values less than 5.
- The range of [tex]\( g(x) \)[/tex] (polynomial function) is [tex]\( (-\infty, \infty) \)[/tex], which includes all real numbers.
In summary, while both functions share the same domain of all real numbers, their ranges differ: [tex]\( f(x) \)[/tex] is restricted to values less than 5, while [tex]\( g(x) \)[/tex] includes all real numbers.
