Find the inverse of the function: `f(x)={(x^3-1, ,x<2), (x^2+3,xgeq2)}`

Answer

`f^(-1)(x)={((x+1)^(3)",",x lt 7),((x-3)^(1//2)",",x ge 7):}`

Explanation

`f(x)={(x^(3)-1",",x lt 2),(x^(2)+3 ",",x ge 2):}`
For `f(x)=x^(3)-1,x lt 2, f^(-1)(x)=(x+1)^(1//3),x lt 7 " "("As " x lt 2 impliesx^(3) lt 8impliesx^(3)-1 lt 7)`
For `f(x)=x^(2)+3, x ge 2, f^(-1)(x)=(x-3)^(1//2),x ge 7 " " ( "As " x ge 2 impliesx^(2) ge 4 implies x^(2)+3 ge 7)`
Hence, `f^(-1)(x)={((x+1)^(3)",",x lt 7),((x-3)^(1//2)",",x ge 7):}`