Question
The domain of the logarithmic function f(x)=lnx is x > 0. Which statement is true about its corresponding exponential function g(x)=e
x
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A. Function g(x) has a range of y≤0. B. Function g(x) has a domain of x≤0. C. Function g(x) has a domain of x > 0. D. Function g(x) has a range of y > 0.
x
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A. Function g(x) has a range of y≤0. B. Function g(x) has a domain of x≤0. C. Function g(x) has a domain of x > 0. D. Function g(x) has a range of y > 0.
Answers
GPT-4o mini
To analyze the exponential function \( g(x) = e^x \), we can look at its domain and range.
1. **Domain of \( g(x) = e^x \)**: The exponential function is defined for all real numbers, so its domain is \( (-\infty, \infty) \). Thus, option B, which states that the domain is \( x \leq 0 \), is false.
2. **Range of \( g(x) = e^x \)**: The output of the exponential function is always positive, as \( e^x > 0 \) for all \( x \). Therefore, the range is \( (0, \infty) \), which means it does not include zero. Thus, option A, which states that the range is \( y \leq 0 \), is also false.
Now, looking at the correct options:
- **C. Function g(x) has a domain of \( x > 0 \)**: This is incorrect because \( g(x) = e^x \) is defined for all real \( x \).
- **D. Function g(x) has a range of \( y > 0 \)**: This is true because \( e^x \) is always greater than zero.
Thus, the correct statement regarding the exponential function \( g(x) = e^x \) is:
**D. Function g(x) has a range of \( y > 0 \).**
1. **Domain of \( g(x) = e^x \)**: The exponential function is defined for all real numbers, so its domain is \( (-\infty, \infty) \). Thus, option B, which states that the domain is \( x \leq 0 \), is false.
2. **Range of \( g(x) = e^x \)**: The output of the exponential function is always positive, as \( e^x > 0 \) for all \( x \). Therefore, the range is \( (0, \infty) \), which means it does not include zero. Thus, option A, which states that the range is \( y \leq 0 \), is also false.
Now, looking at the correct options:
- **C. Function g(x) has a domain of \( x > 0 \)**: This is incorrect because \( g(x) = e^x \) is defined for all real \( x \).
- **D. Function g(x) has a range of \( y > 0 \)**: This is true because \( e^x \) is always greater than zero.
Thus, the correct statement regarding the exponential function \( g(x) = e^x \) is:
**D. Function g(x) has a range of \( y > 0 \).**