Question

Which statement could be used to explain why the function

f(x)=x has an inverse relation that is also a function?

a) The graph of f(x) passes the vertical line test.
b) The graph of the inverse of f(x) is a vertical line.
c) The graph of the inverse of f(x) passes the horizontal line test.
d) The graph of h(c) passes the horizontal line test.

Answer 1

The function f(x)=x has an inverse that is a function because the graph of the inverse passes the horizontal line test, indicating it is one-to-one.

Step-by-step explanation:

The function f(x)=x has an inverse relation that is also a function, and the correct statement to explain this is (c) The graph of the inverse of f(x) passes the horizontal line test. In more detail, for a function to have an inverse that is also a function, each output from the original function must come from exactly one input. Since the function f(x)=x is a one-to-one relationship where each x-value corresponds to one unique y-value, its graph passes the vertical line test, indicating it is a function. Additionally, the inverse of this graph will pass the horizontal line test, confirming that its inverse is also a function.

The inverse of f(x)=x is also f-1(x)=x, which is a reflection of the original function over the line y=x. The graph of the inverse will still be a straight line with a positive slope of 1, passing through the origin, and will also pass both vertical and horizontal line tests, maintaining its status as a function.