Question

The function f(x)=(x-1)^2-4 is not one-to-one. If you restrict the domain for f(x) to x(less than or equal to symbol)1, what is its inverse function and the domain for the inverse?

Answer 1

`\text{We have been given the function }\\ f(x)=(x-1)^2-4\\ \text{Let f(x)=y}\\ y=(x-1)^2-4\\ \text{Swap x and y, we get}\\ \\ x=(y-1)^2-4\\ \text{Now we solve for y}\\ (y-1)^2=x+4\\ \\ y-1=\pm √(x+4)\\ \\ y=1\pm √(x+4)\\ \\ f^(-1)(x)=1\pm √(x+4)\\`

`x+4\geq 0\\ x\geq -4\\ \text{Thus, the domain is given by}\\ x\in [-4,\infty )`