Question

Determine whether each pair of functions \(f\) and \(g\) are inverses. Explain your reasoning. $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{|l|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & -2 & 1 & 4 & 7 & 10 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|} \hline x & -2 & 1 & 4 & 7 & 10 \\ \hline g(x) & -2 & -1 & 0 & 1 & 2 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & 8 & 6 & 4 & 2 & 0 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline g(x) & -8 & -6 & -4 & -2 & 0 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { c. }\\\ &\begin{array}{|l|c|c|c|c|c|} \hline x & -4 & -2 & 0 & 2 & 4 \\ \hline f(x) & 2 & 10 & 18 & 26 & 34 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|} \hline x & -4 & -2 & 0 & 2 & 4 \\ \hline g(x) & \frac{1}{2} & \frac{1}{10} & \frac{1}{18} & \frac{1}{26} & \frac{1}{34} \\ \hline \end{array} \end{aligned} $$

Short answer

Functions f and g in pair a are inverses of each other; functions in pairs b and c are not inverses of each other.

Step-by-step solution

Pair a. Check if \(g(f(x)) = x\)

The pairs (x, f(x)) from the first table for pair a are (-2, -2), (-1, 1), (0, 4), (1, 7), (2, 10). Plug these f(x) values into g, we get that g(-2) = -2, g(1) = -1, g(4) = 0, g(7) = 1, g(10) = 2, which are the x values fed into f, hence for all x in our domain \(g(f(x)) = x\).

Pair a. Check if \(f(g(x)) = x\)

The pairs (x, g(x)) from the second table for pair a are (-2, -2), (1, -1), (4, 0), (7, 1), (10, 2). Plug these g(x) values into f and they yield the x values, hence for all x in our domain \(f(g(x)) = x\). Therefore, f and g are inverses of each other in pair a.

Pair b. Check if \(g(f(x)) = x\)

The pairs (x, f(x)) from the first table for pair b are (2, 8), (3, 6), (4, 4), (5, 2), (6, 0). Plug these f(x) values into g and they yields the negative of the original x values, which don't match. Hence \(g(f(x)) \neq x\) which means the functions f and g in pair b are not inverses of each other.

Pair c. Check if \(g(f(x)) = x\)

The pairs (x, f(x)) from the first table for pair c are (-4, 2), (-2, 10), (0, 18), (2, 26), (4, 34). Plug these f(x) values into g and they yield values that do not correspond to the x values from the f(x) table, so \(g(f(x)) \neq x\). This means the functions f and g in pair c are not inverses of each other.

Key concepts

Function Composition

Function composition is a process to combine two functions where the output of one function becomes the input of another. You can think of it as a machine that processes inputs through two stages to get the final output.
For functions \(f\) and \(g\), the composition \(g(f(x))\) means you first apply \(f\) to \(x\), and then apply \(g\) to the result of \(f(x)\).

• For a valid inverse relation between \(f\) and \(g\), the compositions \(g(f(x)) = x\) and \(f(g(x)) = x\) must hold true for every \(x\) in the domain.
• This means composing \(f\) and \(g\) should essentially "cancel out" the transformation, returning \(x\) to its original state.

In simpler terms, if you undo \(f\) with \(g\), you should end up with what you started with, and vice versa. This test was applied to the functions in the exercise to determine if they were inverses.

Domain and Range

In the context of functions, the domain is the set of all possible input values (\(x\) values) for which the function is defined, while the range is the set of possible output values (\(f(x)\) or \(g(x)\) values).
Understanding the domain and range is crucial when dealing with inverse functions:

• If \(f\) and \(g\) are truly inverse functions, the domain of \(f\) will match the range of \(g\), and vice versa.
• This property is key in checking if the compositions \(g(f(x))\) and \(f(g(x))\) result in the identity function, which would mean returning to the original \(x\).

In the provided example tables, you could evaluate whether each pair of functions meets the criteria by simply checking these "matches" between domains and ranges.

Table Representation of Functions

Tables are a straightforward way to present functions and their values for specific inputs. They effectively illustrate the relationship between inputs and outputs.

• Each row in a table typically corresponds to a specific input-output pair, giving you a concrete understanding of how each input is transformed by the function.
• In the context of checking for inverse pairs, the tables allow you to directly verify if \(f\) and \(g\) undo each other's operations.

Using tables, the problem-solving process is clear: you plug the outputs from one function into the other and see if the original input re-emerges. This method was used extensively in verifying the inverse relationship in the provided solutions.