Answer :
To analyze the inverse function [tex]\( f^{-1}(x) = 1 - 3^{-(y-4)} \)[/tex], let's go through the required aspects step-by-step.
1. Horizontal Asymptote:
To find the horizontal asymptote for the function [tex]\( f^{-1}(x) \)[/tex], we need to determine its behavior as [tex]\( y \)[/tex] approaches infinity.
- As [tex]\( y \to \infty \)[/tex], the term [tex]\( 3^{-(y-4)} \)[/tex] approaches 0 because the exponential term's power becomes a large negative number.
- Thus, [tex]\( 1 - 3^{-(y-4)} \)[/tex] approaches 1 as [tex]\( y \to \infty \)[/tex].
Therefore, the horizontal asymptote of the function is [tex]\( y = 1 \)[/tex].
So the horizontal asymptote is [tex]\( 1 \)[/tex].
2. Range of the Function:
The range of a function is the set of all possible output values. For [tex]\( f^{-1}(x) = 1 - 3^{-(y-4)} \)[/tex]:
- The term [tex]\( 3^{-(y-4)} \)[/tex] is always positive and can be any positive value.
- Since [tex]\( 1 - 3^{-(y-4)} \)[/tex] can take any value less than 1 as [tex]\( 3^{-(y-4)} \)[/tex] can be arbitrarily small but positive, the output of the function can approach but not reach 1.
Hence, the range of the function is [tex]\( (-\infty, 1) \)[/tex].
3. Monotonicity:
To determine whether the function is increasing or decreasing, we examine how the output changes as the input increases.
- As [tex]\( y \)[/tex] increases, [tex]\( 3^{-(y-4)} \)[/tex] decreases, making the overall term [tex]\( 3^{-(y-4)} \)[/tex] smaller.
- Since [tex]\( f^{-1}(x) = 1 - 3^{-(y-4)} \)[/tex], a decrease in [tex]\( 3^{-(y-4)} \)[/tex] leads to an increase in [tex]\( f^{-1}(x) \)[/tex].
Therefore, the function is increasing.
4. Domain of the Function:
The domain of a function consists of all possible input values. For [tex]\( f^{-1}(x) \)[/tex]:
- The expression [tex]\( 1 - 3^{-(y-4)} \)[/tex] can accept any real number as its input because there are no restrictions such as division by zero or square root of a negative number.
Thus, the domain of the function is all real numbers.
So, the solution can be summarized as follows:
- Horizontal asymptote: [tex]\( 1 \)[/tex]
- Range of the function: [tex]\( (-\infty, 1) \)[/tex]
- Monotonicity: Increasing
- Domain of the function: [tex]\( (-\infty, \infty) \)[/tex]
These details fill in the blanks accurately:
The inverse can be given by the function [tex]\( f^{-1}(x)=1-3^{-(y-4)} \)[/tex]. It has a horizontal asymptote of [tex]\( 1 \)[/tex]. The range of the function is [tex]\( (-\infty, 1) \)[/tex], and it is [tex]\( \text{increasing} \)[/tex] on its domain of [tex]\( (-\infty, \infty) \)[/tex].
1. Horizontal Asymptote:
To find the horizontal asymptote for the function [tex]\( f^{-1}(x) \)[/tex], we need to determine its behavior as [tex]\( y \)[/tex] approaches infinity.
- As [tex]\( y \to \infty \)[/tex], the term [tex]\( 3^{-(y-4)} \)[/tex] approaches 0 because the exponential term's power becomes a large negative number.
- Thus, [tex]\( 1 - 3^{-(y-4)} \)[/tex] approaches 1 as [tex]\( y \to \infty \)[/tex].
Therefore, the horizontal asymptote of the function is [tex]\( y = 1 \)[/tex].
So the horizontal asymptote is [tex]\( 1 \)[/tex].
2. Range of the Function:
The range of a function is the set of all possible output values. For [tex]\( f^{-1}(x) = 1 - 3^{-(y-4)} \)[/tex]:
- The term [tex]\( 3^{-(y-4)} \)[/tex] is always positive and can be any positive value.
- Since [tex]\( 1 - 3^{-(y-4)} \)[/tex] can take any value less than 1 as [tex]\( 3^{-(y-4)} \)[/tex] can be arbitrarily small but positive, the output of the function can approach but not reach 1.
Hence, the range of the function is [tex]\( (-\infty, 1) \)[/tex].
3. Monotonicity:
To determine whether the function is increasing or decreasing, we examine how the output changes as the input increases.
- As [tex]\( y \)[/tex] increases, [tex]\( 3^{-(y-4)} \)[/tex] decreases, making the overall term [tex]\( 3^{-(y-4)} \)[/tex] smaller.
- Since [tex]\( f^{-1}(x) = 1 - 3^{-(y-4)} \)[/tex], a decrease in [tex]\( 3^{-(y-4)} \)[/tex] leads to an increase in [tex]\( f^{-1}(x) \)[/tex].
Therefore, the function is increasing.
4. Domain of the Function:
The domain of a function consists of all possible input values. For [tex]\( f^{-1}(x) \)[/tex]:
- The expression [tex]\( 1 - 3^{-(y-4)} \)[/tex] can accept any real number as its input because there are no restrictions such as division by zero or square root of a negative number.
Thus, the domain of the function is all real numbers.
So, the solution can be summarized as follows:
- Horizontal asymptote: [tex]\( 1 \)[/tex]
- Range of the function: [tex]\( (-\infty, 1) \)[/tex]
- Monotonicity: Increasing
- Domain of the function: [tex]\( (-\infty, \infty) \)[/tex]
These details fill in the blanks accurately:
The inverse can be given by the function [tex]\( f^{-1}(x)=1-3^{-(y-4)} \)[/tex]. It has a horizontal asymptote of [tex]\( 1 \)[/tex]. The range of the function is [tex]\( (-\infty, 1) \)[/tex], and it is [tex]\( \text{increasing} \)[/tex] on its domain of [tex]\( (-\infty, \infty) \)[/tex].
