A logarithmic child function has domain of (-2, ∞). What is the range of the exponential child function that is its inverse?
A.(-2, ∞)
B.(2, ∞)
C.(-∞, -2)
D.(-∞, 2)

To solve this problem, let's break it down step by step:

1. **Understanding the logarithmic function**: The given logarithmic child function has a domain of \((-2, ∞)\). This means that the input values of this logarithmic function are from \(-2\) to positive infinity, excluding \(-2\).

2. **Inverse of a function**: The inverse of a function essentially "reverses" the roles of the domain and range. For example, if the logarithmic function has a domain of \((-2, ∞)\), then its range will be the domain of its inverse, the exponential function.

3. **Range of the exponential function**: Since the logarithmic function has a domain of \((-2, ∞)\), its range must be \((-∞, -2)\), because the inverse function will swap the domain and range. Therefore, the range of the exponential function, which is the inverse, will be the same as the range of the original logarithmic function.

Thus, the range of the exponential function is \((-∞, -2)\).

The correct answer is:
**C. \((-∞, -2)\)**.

Would you like more details or have any further questions?

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Here are 5 related questions to expand your understanding:

1. What is the range of a logarithmic function when its domain is positive real numbers?
2. How do you find the inverse of a function?
3. What is the relationship between exponential and logarithmic functions?
4. How does the range of an exponential function change with different base values?
5. Can you explain how the transformations of logarithmic functions affect their domains?

**Tip**: Remember, logarithmic and exponential functions are inverses of each other, meaning they "undo" each other’s effects.

A logarithmic child function has domain of (-2, ∞). What is the range of the exponential child function that is its inverse? A.(-2, ∞) B.(2, ∞) C.(-∞, -2) D.(-∞, 2)

Discover how to find the range of an exponential function that is the inverse of a logarithmic function with domain (-2, ∞). This problem highlights key concepts in inverse functions, logarithms, and exponentials, suitable for high school students in Grades 10-12.

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