Summary
Highlights
A function is invertible if and only if it is a one-to-one function. The horizontal line test is used to determine if a function is one-to-one: if any horizontal line intersects the graph at more than one point, the function is not one-to-one and therefore not invertible.
Graph A fails the horizontal line test because a horizontal line can intersect the function at two points (x1, y1 and x2, y1). This means the function is not one-to-one and thus not invertible. An inverse function would require y1 to be associated with a single x-value, which is not the case here.
Graph B passes the horizontal line test, as any horizontal line intersects the graph at most once. This indicates that the function in Graph B is one-to-one and therefore invertible.
Given an invertible function f with a table of values, to find f inverse of 1, we need to find what input for f results in an output of 1. If f(5) = 1, then f inverse(1) = 5. This is because the inverse function reverses the mapping of the original function.
To find the input for f inverse that results in an output of 1 (f inverse(?) = 1), we look at the original function f. If f inverse outputs 1, then f must have taken 1 as an input. From the table, if f(1) = 9, then the input to f inverse is 9 when its output is 1. So, f inverse(9) = 1.