To solve the given problem, we need to interpret the data provided and find the required function values for $g(t)$ and its inverse $g^{-1}(t)$.

Given Data:

We are provided with pairs of values, where the first number in each pair represents $t$ (input) and the second number represents $g(t)$ (output). The pairs are as follows:

• $g(14) = 83$
• $g(23) = 94$
• $g(24) = 37$
• $g(28) = 108$
• $g(29) = 32$
• $g(34) = 76$
• $g(52) = 82$
• $g(61) = 68$
• $g(102) = 75$
• $g(112) = 61$

1. Finding $g(61)$:

We are directly given that $g(61) = 68$.

2. Finding $g^{-1}(82)$:

To find $g^{-1}(82)$, we need to find the value of $t$ such that $g(t) = 82$. From the table, we see that:

• $g(52) = 82$

Thus, $g^{-1}(82) = 52$.

3. Finding $f(5)$ and $f^{-1}(-4)$:

You mentioned $f(5)$ and $f^{-1}(-4)$, but the provided data only involves $g(t)$ and its inverse. If there is a specific function $f$, that information isn't given in this table, so we'd need more context to solve for $f(5)$ and $f^{-1}(-4)$.

If you provide the definition of $f(t)$, I can continue solving that part.

Summary of Results:

• $g(61) = 68$
• $g^{-1}(82) = 52$

Would you like further details on any of these steps? Here are some related questions:

1. How would you plot the function $g(t)$ based on the given data?
2. What is the process of determining the inverse of a function from a table?
3. How do you determine $g(t)$ if it's not explicitly listed in the table?
4. What are the properties of inverse functions?
5. How can we extend this analysis if more data points are available?

Tip: When finding the inverse from a table, always switch the roles of the input and output values to trace back the original input from the output.