Question

Find an expression for ( g⁻1(x) ), stating its domain and range.

a) ( g⁻1(x) = (√x - 1)² ), domain: ( x ≥ -1 ), range: ( y ≥ 0 )
b) ( g⁻1(x) = (√x + 1)² ), domain: ( x ≥ 0 ), range: ( y ≥ 1 )
c) ( g⁻1(x) = (√x - 1) ), domain: ( x ≥ 1 ), range: ( y ≥ 0 )
d) ( g⁻1(x) = (√x + 1) ), domain: ( x ≥ 0 ), range: ( y ≥ -1 )

Answer 1

To find the inverse function of g, we switch the variables x and y in the original function g(x) and solve for y. The expressions for g⁻¹(x) are: a) (√x - 1)², b) (√x + 1)², c) (√x - 1), and d) (√x + 1). The respective domains and ranges are given.

Step-by-step explanation:

In this question, we are asked to find an expression for the inverse function of g, denoted as g⁻¹(x), and state its domain and range.

To find the inverse function, we need to switch the variables x and y in the original function g(x) and solve for y.

After substituting x with y and solving for y, we get the following expressions for g⁻¹(x):

a) g⁻¹(x) = (√x - 1)², with a domain of x ≥ -1 and a range of y ≥ 0.

b) g⁻¹(x) = (√x + 1)², with a domain of x ≥ 0 and a range of y ≥ 1.

c) g⁻¹(x) = (√x - 1), with a domain of x ≥ 1 and a range of y ≥ 0.

d) g⁻¹(x) = (√x + 1), with a domain of x ≥ 0 and a range of y ≥ -1.