table for g(t) 14,83 23,94 24,37 28,108 29,32 34,76 52,82 61,68 102,75 112, 61 determine f(5) f^-1(-4) g(61) g^-1(82)

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.

table for g(t) 14,83 23,94 24,37 28,108 29,32 34,76 52,82 61,68 102,75 112,61 determine f(5) f^-1(-4) g(61) g^-1(82)

Solve the problem using function values from a table for g(t) and find g(61), g^-1(82). This step-by-step solution walks you through understanding function tables, inverse functions, and related concepts. Suitable for high school students (Grades 9-12).

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