Question
Luis draws this graph of the exponential function f(x)=12(5)x−3. Which inequality shows the range of Luis’s function?
(1 point)
Responses
−∞<y<∞
negative infinity less than y less than infinity
y<−3
y less than negative 3
−3<y<10
negative 3 less than y less than 10
y>−3
(1 point)
Responses
−∞<y<∞
negative infinity less than y less than infinity
y<−3
y less than negative 3
−3<y<10
negative 3 less than y less than 10
y>−3
Answers
GPT-4o mini
To determine the range of the exponential function \( f(x) = 12(5)^{x-3} \), let's analyze the behavior of the function:
1. **Exponential Functions**: The general shape of an exponential function like \( a(b)^{x} \) where \( a > 0 \) and \( b > 0 \) is that it approaches zero but does not actually reach it. This means the function will have a horizontal asymptote.
2. **Transformation**: The given function \( f(x) = 12(5)^{x-3} \) involves a vertical scaling by 12 and a horizontal shift based on \( x - 3 \). The base 5 means it grows large very quickly as \( x \) increases.
3. **Asymptote**: The horizontal asymptote occurs when the exponential part approaches zero. Hence, the lowest the function can get is close to zero, shifted by the initial function’s structure.
4. **Vertical Shift**: In its current form, since we have also a `−3` in the exponent, we need to evaluate it in the range of the outcome due to the scaling factor of 12. The base \( (5)^{x-3} \) approaches zero, but the actual function \( f(x) \) approaches \( 0 + 0 = 0 \).
Thus, we find that:
- The minimum value that \( f(x) \) can approach is 0.
- This means the range of the function is \( f(x) > 0 \).
Given the response options, the appropriate inequality indicating the range of Luis's function is:
**y > −3**
However, if we are only considering the lowest point as approaching a positive zero (from our analysis), we may need the detailed value for just for \( y > 0 \).
So ultimately, the correct interpretation rests likely in:
**y > -3**.
As it suggests that the function will always yield results above, even as it nears down towards positive outcome scaling values.
1. **Exponential Functions**: The general shape of an exponential function like \( a(b)^{x} \) where \( a > 0 \) and \( b > 0 \) is that it approaches zero but does not actually reach it. This means the function will have a horizontal asymptote.
2. **Transformation**: The given function \( f(x) = 12(5)^{x-3} \) involves a vertical scaling by 12 and a horizontal shift based on \( x - 3 \). The base 5 means it grows large very quickly as \( x \) increases.
3. **Asymptote**: The horizontal asymptote occurs when the exponential part approaches zero. Hence, the lowest the function can get is close to zero, shifted by the initial function’s structure.
4. **Vertical Shift**: In its current form, since we have also a `−3` in the exponent, we need to evaluate it in the range of the outcome due to the scaling factor of 12. The base \( (5)^{x-3} \) approaches zero, but the actual function \( f(x) \) approaches \( 0 + 0 = 0 \).
Thus, we find that:
- The minimum value that \( f(x) \) can approach is 0.
- This means the range of the function is \( f(x) > 0 \).
Given the response options, the appropriate inequality indicating the range of Luis's function is:
**y > −3**
However, if we are only considering the lowest point as approaching a positive zero (from our analysis), we may need the detailed value for just for \( y > 0 \).
So ultimately, the correct interpretation rests likely in:
**y > -3**.
As it suggests that the function will always yield results above, even as it nears down towards positive outcome scaling values.