Question

given the function ( f(x) = (x + 11)^3 ), complete parts a through c.
(a) find an equation for ( f^{-1}(x) ).
( f^{-1}(x) = sqrt3{x} - 11 ) (type an exact answer, using radicals as needed.)
(b) choose the correct graph which shows ( f ) and ( f^{-1} ) graphed in the same coordinate system.
(options: graphs labeled a, b, c, d with grids and curves/lines)
(c) state the domain and range of ( f ) and ( f^{-1} ) using interval notation.
the domain of ( f(x) ) is ( (-infty, infty) ), and the range of ( f(x) ) is ( (-infty, infty) ).
the domain of ( f^{-1}(x) ) is ( (-infty, infty) ), and the range of ( f^{-1}(x) ) is ( (-infty, infty) ).

Explanation:

Step1: Find the inverse function

Given \( f(x)=(x + 11)^3 \). To find the inverse, we first let \( y=(x + 11)^3 \). Then we solve for \( x \) in terms of \( y \). Take the cube - root of both sides: \( \sqrt[3]{y}=x + 11 \). Then \( x=\sqrt[3]{y}-11 \). Swap \( x \) and \( y \) to get the inverse function \( f^{-1}(x)=\sqrt[3]{x}-11 \).

Step2: Analyze the graphs of \( f(x) \) and \( f^{-1}(x) \)

The function \( f(x)=(x + 11)^3 \) is a cubic function. The graph of \( y = x^3 \) is a cubic curve passing through the origin, and \( f(x)=(x + 11)^3 \) is a horizontal shift of \( y=x^3 \) 11 units to the left. The inverse function \( f^{-1}(x)=\sqrt[3]{x}-11 \) is a horizontal shift of \( y = \sqrt[3]{x} \) 11 units down. Also, the graph of a function and its inverse are symmetric about the line \( y = x \).

Looking at the options:

• Option A: The two graphs do not seem to be symmetric about \( y=x \).
• Option B: The graph of \( f(x)=(x + 11)^3 \) (the non - square - root, non - shifted - down - cubic - like graph) and \( f^{-1}(x)=\sqrt[3]{x}-11 \) (the square - root - like graph shifted down) are symmetric about \( y = x \) (the dashed line). The shape of \( f(x)=(x + 11)^3 \) is a cubic curve, and \( f^{-1}(x)=\sqrt[3]{x}-11 \) is a cube - root curve, and their symmetry about \( y = x \) is evident here.
• Option C: The two graphs do not show the correct symmetry or the correct shapes for \( f(x) \) and \( f^{-1}(x) \).
• Option D: The two graphs do not have the correct shapes or symmetry for the given function and its inverse.

Answer:

The correct graph is Option B.