Given the function [tex]f(x)=x^3-6[/tex], complete parts a through c.
(a) Find an equation for [tex]f^{-1}(x)[/tex].
(b) Graph f and f[tex]{ }^{-1}[/tex] in the same rectangular coordinate system.
(c) Use interval notation to give the domain and the range of f and [tex]f^{-1}[/tex]
(a) Find [tex]f ^{-1}( x )[/tex]
[tex]f^{-1}(x)=\sqrt[3]{x+6}[/tex]
(Type an exact answer, using radicals as needed.)
(b) Graph f and f [tex]{ }^{-1}[/tex] in the same coordinate system. Choose the correct graph below.
A.
B.
C.
D.
Answer (2)
Find the inverse function by swapping x and y and solving for y : f − 1 ( x ) = 3 x + 6 .
Determine the domain and range of f ( x ) = x 3 − 6 : Domain is ( − ∞ , ∞ ) , Range is ( − ∞ , ∞ ) .
Determine the domain and range of f − 1 ( x ) = 3 x + 6 : Domain is ( − ∞ , ∞ ) , Range is ( − ∞ , ∞ ) .
Identify the correct graph representing f ( x ) and f − 1 ( x ) , which is graph C.
C
Explanation
Problem Analysis We are given the function f ( x ) = x 3 − 6 and asked to find its inverse, graph both functions, and determine their domains and ranges.
Finding the Inverse Function To find the inverse function, we swap x and y in the equation y = x 3 − 6 and solve for y . This gives us x = y 3 − 6 . Adding 6 to both sides, we get x + 6 = y 3 . Taking the cube root of both sides, we find y = 3 x + 6 . Thus, f − 1 ( x ) = 3 x + 6 .
Domain and Range of f(x) The function f ( x ) = x 3 − 6 is a cubic function, which is defined for all real numbers. Therefore, its domain is ( − ∞ , ∞ ) . Since it's a cubic function, it also takes on all real values, so its range is ( − ∞ , ∞ ) .
Domain and Range of f^{-1}(x) The inverse function f − 1 ( x ) = 3 x + 6 is a cube root function, which is also defined for all real numbers. Therefore, its domain is ( − ∞ , ∞ ) . Similarly, it takes on all real values, so its range is ( − ∞ , ∞ ) .
Graphing the Functions Now, let's consider the graphs. The graph of f ( x ) = x 3 − 6 is a cubic function shifted down by 6 units. The graph of f − 1 ( x ) = 3 x + 6 is a cube root function shifted left by 6 units. The graph of the inverse function is a reflection of the original function across the line y = x . Looking at the options, graph C correctly represents these functions.
Final Answer In summary:
f − 1 ( x ) = 3 x + 6
The domain of f ( x ) is ( − ∞ , ∞ ) , and its range is ( − ∞ , ∞ ) .
The domain of f − 1 ( x ) is ( − ∞ , ∞ ) , and its range is ( − ∞ , ∞ ) .
The correct graph is C.
Examples
Imagine you are designing a water tank. The volume of water in the tank is given by V = s 3 − 6 , where s is the side length of the tank. If you want to find the side length required for a specific volume, you need to use the inverse function s = 3 V + 6 . This allows you to determine the dimensions of the tank based on the desired volume. Understanding inverse functions is crucial in many engineering and design applications where you need to reverse a relationship to find an input for a desired output.
The inverse function of f ( x ) = x 3 − 6 is f − 1 ( x ) = 3 x + 6 . Both functions have domains and ranges of ( − ∞ , ∞ ) . The correct graph depicting both functions is option C.
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