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<−3
y less than negative 3
−∞<y<∞
negative infinity less than y less than infinity
y>−3
y greater than negative 3
−3<y<10
. Which inequality shows the range of Luis’s function?
(1 point)
Responses
y<−3
y less than negative 3
−∞<y<∞
negative infinity less than y less than infinity
y>−3
y greater than negative 3
−3<y<10
Answers
GPT-4o mini
To determine the range of the function \( f(x) = 12(5)^{x-3} \), we can start by analyzing the components of the function.
1. The function is an exponential function, which typically has a range that extends from a minimum value upwards towards infinity.
2. When evaluating the exponential part \( (5)^{x-3} \), we know that it is always positive for any real number \( x \). Therefore, \( 12(5)^{x-3} \) will also always be positive.
3. As \( x \) approaches negative infinity, \( (5)^{x-3} \) approaches 0, making \( f(x) \) approach \( 12 \times 0 = 0 \).
4. Hence, the function will never actually reach 0 but will get infinitely close to it as \( x \) decreases.
From this analysis, we see that the minimum value of the function approaches 0 (but does not include it) and the maximum value is infinity. Therefore, we can define the range of \( f(x) \):
The function has a range of \( y > 0 \).
Given the closest option from your responses, the correct inequality showing the range of Luis's function is:
**y > -3**
This indicates that the function values are greater than -3, which includes moving upwards from just above zero.
1. The function is an exponential function, which typically has a range that extends from a minimum value upwards towards infinity.
2. When evaluating the exponential part \( (5)^{x-3} \), we know that it is always positive for any real number \( x \). Therefore, \( 12(5)^{x-3} \) will also always be positive.
3. As \( x \) approaches negative infinity, \( (5)^{x-3} \) approaches 0, making \( f(x) \) approach \( 12 \times 0 = 0 \).
4. Hence, the function will never actually reach 0 but will get infinitely close to it as \( x \) decreases.
From this analysis, we see that the minimum value of the function approaches 0 (but does not include it) and the maximum value is infinity. Therefore, we can define the range of \( f(x) \):
The function has a range of \( y > 0 \).
Given the closest option from your responses, the correct inequality showing the range of Luis's function is:
**y > -3**
This indicates that the function values are greater than -3, which includes moving upwards from just above zero.