https://file.aiquickdraw.com/mathbot/file/1726784689471rmwvogma.jpg

The image contains a graph of two functions, one represented by a solid line and the other by a dashed line. The question is asking whether these functions are inverses of each other, focusing on logarithmic and exponential functions.

### Explanation:
The functions appear to represent an exponential function and its inverse, a logarithmic function. The inverse of an exponential function is a logarithmic function, and vice versa. Inverse functions reflect over the line \( y = x \), meaning that if you reflect the exponential function over this line, it should result in the logarithmic function.

### Analyzing the choices:

- **A.** Incorrect. Inverse functions do not need to have their intercepts align in any particular way except that they reflect over the line \( y = x \).
  
- **B.** Correct. The key property of inverse functions is that reflecting one over the line \( y = x \) results in the other. This matches the description of exponential and logarithmic functions being inverses.
  
- **C.** Incorrect. This reasoning is not related to inverse functions. The position relative to the x-axis does not determine if functions are inverses.
  
- **D.** Incorrect. The fact that neither passes through the origin does not determine whether the functions are inverses.

### Correct Answer:
**B. The functions are inverses because reflecting the function with a solid line across the line \( y = x \) results in the function with a dashed line.**

Would you like further details or explanations?

### Related Questions:
1. What is the relationship between logarithmic and exponential functions in general?
2. How can you verify whether two functions are inverses graphically?
3. What is the general equation of an exponential function and its inverse logarithmic function?
4. What role does the line \( y = x \) play in determining inverse functions?
5. How can transformations (translations, reflections) affect the graphs of logarithmic and exponential functions?

### Tip:
To check if two functions are inverses, verify if their compositions \( f(g(x)) \) and \( g(f(x)) \) both equal \( x \).

Are the logarithmic and exponential functions shown inverses of each other? Explain.

Explore whether logarithmic and exponential functions are inverses of each other, with an emphasis on graph reflection over the line y = x. This detailed explanation covers inverse functions, their graphical relationships, and the reflection property. Suitable for Grades 10-12 students learning about advanced algebra.

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