Answer :
Answer:
C) [-2, 7]
Step-by-step explanation:
Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
The graph shows a continuous curve f(x) with closed circle endpoints at x = -6 and x = 10. The closed circles indicate that those values of x are included in the domain. Therefore, the domain of the graphed function f(x) is [-6, 10].
[tex]\dotfill[/tex]
Range
The range of a function is the set of all possible output values (y-values) for which the function is defined.
The minimum y-value of the curve is y = -2 and the maximum y-value is y = 7. As both these values are included in the range, the range of the graphed function f(x) is [-2, 7].
[tex]\dotfill[/tex]
Inverse function
An inverse function essentially reverses the operation done by a given function. For a function to have an inverse, it must be bijective, meaning each output is associated with exactly one input, and every possible output is achieved by some input.
Graphically, the inverse function is a reflection of the original function across the line y = x. Therefore:
- The domain of the inverse function is the range of the original function.
- The range of the inverse function is the domain of the original function.
In this case, the graphed function f(x) is bijective, so it has an inverse f⁻¹(x). Therefore:
- The domain of f⁻¹(x) is [-2, 7].
- The range of f⁻¹(x) is [-6, 10].
So, the domain of f⁻¹(x) is:
[tex]\Large\boxed{\boxed{\textsf{Domain of $f^{-1}(x)$}=[-2, 7]}}[/tex]
Finding the Domain and Range from the Graph:
1. Domain of f(x):
- The domain is the set of all possible x-values for which the function f(x) is defined.
- To find the domain, observe the horizontal extent of the graph. Identify the leftmost and rightmost points on the graph along the x-axis.
- In the given graph, the leftmost point appears to be at x = -6 and the rightmost point appears to be at x = 10.
- Thus, the domain of f(x) is [-6, 10].
Note:
- Use square brackets [] when the endpoints are included in the domain (i.e. the graph touches or includes the points). Notice, the end points of the given graph are dark circled, which means the graph includes the points x = -6 and x = 10.
- Use small parentheses ( ) if the endpoints are not included (i.e. the graph approaches but does not touch or include the points). The endpoints of a graph are open circles to indicate that the endpoints are not included.
2. Range of f(x):
- The range is the set of all possible y-values that the function f(x) can take.
- To find the range, observe the vertical extent of the graph. Identify the lowest and highest points on the graph along the y-axis.
- In the given graph, the lowest point appears to be at y = -2 and the highest point appears to be at y = 7.
- Thus , the range of f(x) is [-2, 7].
Note:
- The rule of using the brackets type is the same as the one we will use to find the domain interval.
Finding the Domain of the Inverse Function
- The domain of the inverse function [tex]f^{-1}(x)[/tex] is the same as the range of the original function [tex]f(x).[/tex]
- From the previous step, we found that the range of f(x) is [-2, 7].
- Therefore, the domain of [tex]f^{-1}(x)[/tex] is also [-2, 7].
Conclusion:
Based on the given options, the closest interval to the domain of [tex]f^{-1}(x)[/tex] is:
[tex]\boxed{[-2,7].}[/tex]
