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A table of values is a convenient way to represent how a function behaves at certain points. For each input value, or 't', the table provides the corresponding output value of the function. This is particularly useful when dealing with inverse functions, where identifying specific output values helps us discover the input that led to that output.

The table allows us to work backwards: given an output, we can find the associated input by locating the output in the table and seeing its matching input. This is a great visual tool for solving inverse function problems because it provides a quick reference to identify pairs of input and output.

For example, if you have a table that shows values for the function \( f(t) \), and you need \( f^{-1}(5) \), you can find this by looking for 5 in the row of \( f(t) \) and reading its corresponding \( t \) value. This is precisely how we found that \( f^{-1}(5) = 3 \) in the given exercise.

A one-to-one function, or injective function, is special because it ensures that each element in the function's range is mapped from a unique element in its domain. This unique mapping means that no two different inputs can have the same output, which is crucial for a function to have an inverse.

In simple terms, if a function is one-to-one, each output value comes from one and only one input value. This property of functions matters significantly when determining inverses because only one-to-one functions have inverses that are also functions.

In the exercise, when we looked at the function \( g(t) \), we saw both \( g(2) = 6 \) and \( g(4) = 6 \). Because two different inputs give the same output, \( g(t) \) is not one-to-one. Hence, when a function isn't one-to-one, it can’t have a well-defined inverse that assigns exactly one output (input in the inverse sense) to each potential input.

Function inverses undo the effect of a function. If \( f \) maps \( x \) to \( y \), the inverse \( f^{-1} \) maps \( y \) back to \( x \). This is only possible when the function is one-to-one and onto (bijective), ensuring every output has a corresponding input.

To find the inverse of a function using a table, you look at each output value to find the corresponding input. Suppose \( f(t) = y \), then \( f^{-1}(y) = t \).

Our task was to find \( f^{-1}(5) \) and \( g^{-1}(6) \). We could find \( f^{-1}(5) = 3 \), which means when \( f \) produces 5, it started with input 3. This process of reversing functions is vital for numerous mathematical analyses, such as solving equations and modeling inverses in real-world applications like decoding signals.