Questions: What would be the restrictions of the domain to make this function an invertible function? x is between -1 and 1 , inclusive x is between -5 and 0 , inclusive x is between -1 and 0 , inclusive None of the above.

What would be the restrictions of the domain to make this function an invertible function?
x is between -1 and 1 , inclusive
x is between -5 and 0 , inclusive
x is between -1 and 0 , inclusive
None of the above.

Transcript text: What would be the restrictions of the domain to make this function an invertible function? $x$ is between -1 and 1 , inclusive $x$ is between -5 and 0 , inclusive $x$ is between -1 and 0 , inclusive None of the above.

Solution

Solution Steps

Step 1: Identify the function and its properties

The function $ y = f(x) $ shown in the graph appears to be a rational function with vertical asymptotes and a parabolic shape between the asymptotes. To make this function invertible, it must pass the Horizontal Line Test, meaning any horizontal line should intersect the graph at most once.

Step 2: Analyze the graph for invertibility

The graph has vertical asymptotes at $ x = -1 $ and $ x = 1 $. Between these asymptotes, the function is continuous and strictly decreasing. This interval is a candidate for making the function invertible.

Step 3: Determine the appropriate domain restriction

To ensure the function is invertible, we restrict the domain to the interval where the function is one-to-one. The interval $ x $ between $-1$ and $1$ (inclusive) ensures that the function is strictly decreasing and passes the Horizontal Line Test.