Find the inverse of the given one-to-one function [tex]f[/tex]. Give the domain and the range of [tex]f[/tex] and of [tex]f^{-1}[/tex], and graph both [tex]f[/tex] and [tex]f^{-1}[/tex] on the same set of axes.
[tex]... x+5[/tex]
Choose the correct graphs for [tex]f[/tex] and [tex]f^{-1}[/tex].
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Answer (2)
Find the inverse function by swapping x and y in y = x + 5 and solving for y , resulting in f − 1 ( x ) = x − 5 .
Determine the domain and range of f ( x ) = x + 5 , which are both ( − ∞ , ∞ ) since it's a linear function.
Determine the domain and range of f − 1 ( x ) = x − 5 , which are also both ( − ∞ , ∞ ) since it's a linear function.
Identify the correct graph showing f ( x ) and f − 1 ( x ) as reflections across the line y = x with y-intercepts at 5 and -5, respectively.
The inverse function is f − 1 ( x ) = x − 5 .
Explanation
Problem Analysis The given function is f ( x ) = x + 5 . We need to find its inverse, the domain and range of both the function and its inverse, and identify the correct graph.
Finding the Inverse Function To find the inverse function, we replace f ( x ) with y , so we have y = x + 5 . Then, we swap x and y to get x = y + 5 . Now, we solve for y :
x = y + 5 y = x − 5 So, the inverse function is f − 1 ( x ) = x − 5 .
Domain and Range of f(x) The function f ( x ) = x + 5 is a linear function. Linear functions have a domain and range of all real numbers. Therefore, the domain and range of f ( x ) are ( − ∞ , ∞ ) .
Domain and Range of f^{-1}(x) Similarly, the inverse function f − 1 ( x ) = x − 5 is also a linear function. Thus, its domain and range are also all real numbers, ( − ∞ , ∞ ) .
Analyzing the Graphs Now, let's consider the graphs. The function f ( x ) = x + 5 is a line with a slope of 1 and a y-intercept of 5. The inverse function f − 1 ( x ) = x − 5 is a line with a slope of 1 and a y-intercept of -5. The graph of the inverse function is a reflection of the original function across the line y = x .
Looking at the options, we need to identify the graph that shows both lines with the correct intercepts and the reflection property.
Conclusion Based on the above analysis, the correct graph should show two lines, f ( x ) = x + 5 and f − 1 ( x ) = x − 5 , where f − 1 ( x ) is the reflection of f ( x ) across the line y = x . Without the actual graphs to choose from, I can't definitively pick one. However, the key features to look for are the y-intercepts at 5 and -5, and the lines should appear to be reflections of each other across the line y = x .
Examples
Understanding inverse functions is crucial in many areas of mathematics and real-world applications. For instance, if you have a function that converts Celsius to Fahrenheit, the inverse function converts Fahrenheit back to Celsius. Similarly, in cryptography, encryption and decryption are inverse functions of each other, ensuring secure communication. The ability to find and understand inverse functions allows us to reverse processes and solve problems in various fields, from science and engineering to economics and computer science.
The inverse of the function f ( x ) = x + 5 is f − 1 ( x ) = x − 5 . Both functions have a domain and range of ( − ∞ , ∞ ) . The correct graphs will reflect each other across the line y = x , with f ( x ) having a y-intercept at 5 and f − 1 ( x ) at -5.
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