Question

Which of the following linear functions would not have an inverse that is also a function? Explain how you made your choice.

a) y = x
b) y = 2
c) 2y = x
d) y = 5x - 1

Answer 1

Function b) y = 2 would not have an inverse that is also a function because it is a horizontal line that fails the Horizontal Line Test, indicating that it does not have a one-to-one relationship.

Step-by-step explanation:

The given functions are being evaluated to determine which one would not have an inverse that is also a function. The key principle to be used here is the Horizontal Line Test. A function has an inverse that is also a function if and only if any horizontal line passes through the graph at most once.

• a) y = x: This equation represents a linear function with a slope of 1 and an intercept of 0. It clearly passes the Horizontal Line Test.
• b) y = 2: This equation is a horizontal line, which fails the Horizontal Line Test because a horizontal line would intersect this function at every point, indicating there is not a one-to-one relationship.
• c) 2y = x: After rearrangement, y = x/2, it is still a linear function with a slope that would pass the Horizontal Line Test, similar to example a).
• d) y = 5x - 1: This is a linear function with a slope of 5 and y-intercept of -1, which also passes the Horizontal Line Test.

The correct choice is b) y = 2, because it does not pass the Horizontal Line Test and therefore would not have an inverse that is also a function.