Question

1. for the inverse of y = f(x) to be a function, a restricted domain of y = f(x) is required. circle all of the following restricted domains so that the inverse of y = f(x) is a function.

a) -4,-3 b) -4,-2
c) -3,2 d) -2,1
e) -3,0 f) 1,3

Explanation:

Step1: Recall function - inverse condition

A function \(y = f(x)\) has an inverse that is a function if \(y = f(x)\) is one - to - one on the domain. A one - to - one function passes the horizontal line test (a horizontal line intersects the graph of the function at most once).

Step2: Analyze each interval

• For interval \([-4,-3]\): On the interval \([-4,-3]\), the function \(y = f(x)\) is one - to - one.
• For interval \([-4,-2]\): The function is not one - to - one on \([-4,-2]\) since a horizontal line can intersect the graph more than once in this interval.
• For interval \([-3,2]\): The function is not one - to - one on \([-3,2]\) as a horizontal line can intersect the graph more than once.
• For interval \([-2,1]\): The function is one - to - one on \([-2,1]\).
• For interval \([-3,0]\): The function is not one - to - one on \([-3,0]\) as a horizontal line can intersect the graph more than once.
• For interval \([1,3]\): The function is one - to - one on \([1,3]\).

Answer:

A. \([-4,-3]\), D. \([-2,1]\), F. \([1,3]\)