Question

u3:04: unit 3 test
how can you determine if an inverse is a function?
0 / 10000 word limit

Explanation:

To determine if the inverse of a relation is a function, we can use two main methods related to the original function:

1. Horizontal Line Test on the Original Function: If no horizontal line intersects the graph of the original function more than once, the function is one - to - one. A one - to - one function has an inverse that is also a function. This is because a one - to - one function satisfies the condition that each output is associated with exactly one input, so when we reverse the mapping (to get the inverse), each input of the inverse (which was the output of the original) will be associated with exactly one output of the inverse (which was the input of the original), thus satisfying the definition of a function.

1. Analyzing the Inverse Relation: For the inverse relation (found by swapping the \(x\) - and \(y\) - values of the original function's ordered pairs), we can check if it satisfies the definition of a function. A relation is a function if each input ( \(x\) - value) is associated with exactly one output ( \(y\) - value). So, we examine the inverse relation to see if any \(x\) - value is paired with more than one \(y\) - value. If not, the inverse is a function.

Answer:

To determine if an inverse is a function, we can: 1) Apply the horizontal line test to the original function—if no horizontal line intersects its graph more than once (it is one - to - one), its inverse is a function. 2) Analyze the inverse relation (formed by swapping \(x\) and \(y\) of the original function’s ordered pairs) to ensure each \(x\) - value is paired with exactly one \(y\) - value (satisfying the function definition).