Question
Which statement accurately analyzes a key feature of a logarithmic parent function?
A. The logarithmic parent function is an even function.
B. The logarithmic parent function has a limited domain.
C. The logarithmic parent function has only one y-intercept.
D. The logarithmic parent function can have multiple x-intercepts.
A. The logarithmic parent function is an even function.
B. The logarithmic parent function has a limited domain.
C. The logarithmic parent function has only one y-intercept.
D. The logarithmic parent function can have multiple x-intercepts.
Answers
Answered by
GPT-4o mini
The statement that accurately analyzes a key feature of a logarithmic parent function is:
**C. The logarithmic parent function has only one y-intercept.**
The logarithmic parent function, \( f(x) = \log_b(x) \) (where \( b > 0 \) and \( b \neq 1 \)), has a domain of \( x > 0 \), meaning it does not have an x-intercept when considered on its own. However, the function does intersect the y-axis at the point (1, 0), which means it has only one y-intercept. Therefore, option C is correct.
Option analysis:
- A: The logarithmic function is not an even function; it is neither even nor odd.
- B: While the logarithmic parent function does have a restricted domain (only positive x-values), this statement does not capture the most distinctive feature compared to option C.
- D: The logarithmic parent function has only one x-intercept (which occurs at \( x = 1 \) for the base 10 logarithm and at \( x = e \) for the natural logarithm). Thus, it cannot have multiple x-intercepts.
Thus, C is the best and most accurate analysis.
**C. The logarithmic parent function has only one y-intercept.**
The logarithmic parent function, \( f(x) = \log_b(x) \) (where \( b > 0 \) and \( b \neq 1 \)), has a domain of \( x > 0 \), meaning it does not have an x-intercept when considered on its own. However, the function does intersect the y-axis at the point (1, 0), which means it has only one y-intercept. Therefore, option C is correct.
Option analysis:
- A: The logarithmic function is not an even function; it is neither even nor odd.
- B: While the logarithmic parent function does have a restricted domain (only positive x-values), this statement does not capture the most distinctive feature compared to option C.
- D: The logarithmic parent function has only one x-intercept (which occurs at \( x = 1 \) for the base 10 logarithm and at \( x = e \) for the natural logarithm). Thus, it cannot have multiple x-intercepts.
Thus, C is the best and most accurate analysis.