Question
Stephanie is given a graph that has no maximum value, is always increasing, and is not negative. Based on these behaviors, which of the following options is the parent function?
Answer
Multiple Choice Answers
f, of, x, equals, log, xf(x)=logx
f, of, x, equals, 3, to the power xf(x)=3
x
f, of, x, equals, xf(x)=x
f, of, x, equals, x, squaredf(x)=x
2
Answer
Multiple Choice Answers
f, of, x, equals, log, xf(x)=logx
f, of, x, equals, 3, to the power xf(x)=3
x
f, of, x, equals, xf(x)=x
f, of, x, equals, x, squaredf(x)=x
2
Answers
GPT-4o mini
Based on the behaviors described, the correct choice is:
**\( f(x) = 3^x \)**
Here's the reasoning:
1. **No Maximum Value**: The function must grow indefinitely, meaning as \( x \) increases, \( f(x) \) keeps increasing without reaching a maximum.
2. **Always Increasing**: The function must be strictly increasing, which means that as you increase \( x \), \( f(x) \) also increases.
3. **Not Negative**: The function values must always be zero or positive.
Now, let's analyze the options:
- **\( f(x) = \log x \)**: This function is not defined for \( x \leq 0 \) and increases, but it has no maximum value. However, it does not meet the "not negative" criterion as it is negative for \( 0 < x < 1 \).
- **\( f(x) = 3^x \)**: This function is always positive, always increasing, and has no maximum value. As \( x \) increases, \( f(x) \) increases exponentially.
- **\( f(x) = x \)**: This function is indeed always increasing and has no maximum value. However, it meets the "not negative" criterion, but it is strictly increasing and unbounded, which could also be a candidate.
- **\( f(x) = x^2 \)**: This function has a minimum value at \( x = 0 \) and isn’t strictly increasing for all \( x \) (it decreases for \( x < 0 \)).
Based on the overall behaviors described, **both \( f(x) = 3^x \) and \( f(x) = x \)** can be legitimate answers. However, since \( 3^x \) is a clearer example of a function that grows without bound (exponentially), it is the best fit for the description given.
Thus, **the best option is \( f(x) = 3^x \)**.
**\( f(x) = 3^x \)**
Here's the reasoning:
1. **No Maximum Value**: The function must grow indefinitely, meaning as \( x \) increases, \( f(x) \) keeps increasing without reaching a maximum.
2. **Always Increasing**: The function must be strictly increasing, which means that as you increase \( x \), \( f(x) \) also increases.
3. **Not Negative**: The function values must always be zero or positive.
Now, let's analyze the options:
- **\( f(x) = \log x \)**: This function is not defined for \( x \leq 0 \) and increases, but it has no maximum value. However, it does not meet the "not negative" criterion as it is negative for \( 0 < x < 1 \).
- **\( f(x) = 3^x \)**: This function is always positive, always increasing, and has no maximum value. As \( x \) increases, \( f(x) \) increases exponentially.
- **\( f(x) = x \)**: This function is indeed always increasing and has no maximum value. However, it meets the "not negative" criterion, but it is strictly increasing and unbounded, which could also be a candidate.
- **\( f(x) = x^2 \)**: This function has a minimum value at \( x = 0 \) and isn’t strictly increasing for all \( x \) (it decreases for \( x < 0 \)).
Based on the overall behaviors described, **both \( f(x) = 3^x \) and \( f(x) = x \)** can be legitimate answers. However, since \( 3^x \) is a clearer example of a function that grows without bound (exponentially), it is the best fit for the description given.
Thus, **the best option is \( f(x) = 3^x \)**.