We have the function
[tex]f(x)=(x+6)^3[/tex]
1. For f^-1:
Let y = f(x) = (x+6)^3
Switch x and y to get:
[tex]x=(y+6)^3[/tex]
And solve for y
[tex]\begin{gathered} x^{\frac{1}{3}}=y+6 \\ x^{\frac{1}{3}}-6=y+6-6 \\ x^{\frac{1}{3}}-6=y \end{gathered}[/tex]
And we have y = f^-1(x)
Answer blank 1:
[tex]f^{-1}(x)=x^{\frac{1}{3}}-6[/tex]
2. For f o f^-1 (x):
[tex](f\circ f^{-1})(x)=f(f^{-1}(x))[/tex]
And solve
[tex]\begin{gathered} =f(x^{\frac{1}{3}}-6) \\ =(x^{\frac{1}{3}}-6+6)^3 \\ =(x^{\frac{1}{3}})^3 \\ =x \end{gathered}[/tex]
answer blank 2
[tex]x^{\frac{1}{3}}-6[/tex]
answer blank 3
[tex]x^{\frac{1}{3}}-6[/tex]
answer blank 4
[tex]x^{\frac{1}{3}}[/tex]
3. For f^-1 o f:
[tex](f^{-1}\circ f)(x)=f^{-1}(f(x))[/tex]
Solve
[tex]\begin{gathered} =f^{-1}((x+6)^3) \\ =\sqrt[3]{(x+6)^3}-6 \\ =x+6-6 \\ =x \end{gathered}[/tex]
answer blank 5
[tex](x+6)^3[/tex]
answer blank 6
[tex](x+6)^3[/tex]
answer blank 7
[tex]x+6[/tex]
