Question
An exponential function has the equation f(x)=−3^x−2
. Complete the inequality to show the range of the function.
The range of the function is y __
. Complete the inequality to show the range of the function.
The range of the function is y __
Answers
GPT-4o mini
To determine the range of the exponential function \( f(x) = -3^x - 2 \), we start by analyzing the behavior of the function.
1. The basic exponential function \( 3^x \) is always positive and increases rapidly as \( x \) increases. Therefore, \( 3^x \) has a minimum value of 0, which occurs when \( x \) approaches negative infinity.
2. Since \( f(x) \) is defined as \( -3^x - 2 \), we can see how the transformations affect the range:
- The term \( -3^x \) means that instead of being positive, the value will be negative and will approach 0 from below as \( x \) increases.
- By subtracting 2, we shift the entire function downward by 2.
As \( x \) approaches infinity, \( 3^x \) goes to infinity and thus \( -3^x \) approaches negative infinity. Consequently, \( f(x) \) approaches \( -\infty \).
As \( x \) approaches negative infinity, \( 3^x \) approaches 0 (meaning \( -3^x \) approaches 0 from the negative side), and thus \( f(x) \) approaches \( -2 \) from below.
Based on this analysis:
- The maximum value of \( f(x) \) is just below \(-2\) (approaching but never reaching \(-2\)).
- The minimum value of \( f(x) \) is \( -\infty \).
Thus, the range of the function is
\[ y < -2. \]
So, the complete inequality for the range is:
The range of the function is \( y < -2 \).
1. The basic exponential function \( 3^x \) is always positive and increases rapidly as \( x \) increases. Therefore, \( 3^x \) has a minimum value of 0, which occurs when \( x \) approaches negative infinity.
2. Since \( f(x) \) is defined as \( -3^x - 2 \), we can see how the transformations affect the range:
- The term \( -3^x \) means that instead of being positive, the value will be negative and will approach 0 from below as \( x \) increases.
- By subtracting 2, we shift the entire function downward by 2.
As \( x \) approaches infinity, \( 3^x \) goes to infinity and thus \( -3^x \) approaches negative infinity. Consequently, \( f(x) \) approaches \( -\infty \).
As \( x \) approaches negative infinity, \( 3^x \) approaches 0 (meaning \( -3^x \) approaches 0 from the negative side), and thus \( f(x) \) approaches \( -2 \) from below.
Based on this analysis:
- The maximum value of \( f(x) \) is just below \(-2\) (approaching but never reaching \(-2\)).
- The minimum value of \( f(x) \) is \( -\infty \).
Thus, the range of the function is
\[ y < -2. \]
So, the complete inequality for the range is:
The range of the function is \( y < -2 \).