Question
An exponential function has the equation f(x)=−3^x−2
. Complete the inequality to show the range of the function.
. Complete the inequality to show the range of the function.
Answers
GPT-4o mini
To analyze the range of the function \( f(x) = -3^x - 2 \), we start by considering the behavior of the function as \( x \) varies.
1. The term \( 3^x \) is always positive for all real values of \( x \) (since the base 3 is greater than 1).
2. Consequently, \( -3^x \) will be always negative (and approaches 0 as \( x \) approaches negative infinity).
3. Therefore, \( -3^x - 2 \) means we are taking \( -3^x \) and shifting it down by 2 units.
As \( x \) approaches positive infinity, \( 3^x \) approaches infinity, making \( -3^x \) approach negative infinity. This implies:
\[
f(x) \to -\infty \quad \text{as } x \to +\infty.
\]
As \( x \) approaches negative infinity, \( 3^x \) approaches 0, leading to:
\[
f(x) \to -2 \quad \text{as } x \to -\infty.
\]
Thus, \( -3^x - 2 \) will always take values less than -2. The maximum value of the function occurs when \( x \) approaches negative infinity, which is \( -2 \).
Therefore, the range of the function can be expressed in the form of an inequality:
\[
f(x) < -2.
\]
In interval notation, the range is:
\[
(-\infty, -2).
\]
So the completed inequality to show the range of the function is:
\[
f(x) < -2.
\]
1. The term \( 3^x \) is always positive for all real values of \( x \) (since the base 3 is greater than 1).
2. Consequently, \( -3^x \) will be always negative (and approaches 0 as \( x \) approaches negative infinity).
3. Therefore, \( -3^x - 2 \) means we are taking \( -3^x \) and shifting it down by 2 units.
As \( x \) approaches positive infinity, \( 3^x \) approaches infinity, making \( -3^x \) approach negative infinity. This implies:
\[
f(x) \to -\infty \quad \text{as } x \to +\infty.
\]
As \( x \) approaches negative infinity, \( 3^x \) approaches 0, leading to:
\[
f(x) \to -2 \quad \text{as } x \to -\infty.
\]
Thus, \( -3^x - 2 \) will always take values less than -2. The maximum value of the function occurs when \( x \) approaches negative infinity, which is \( -2 \).
Therefore, the range of the function can be expressed in the form of an inequality:
\[
f(x) < -2.
\]
In interval notation, the range is:
\[
(-\infty, -2).
\]
So the completed inequality to show the range of the function is:
\[
f(x) < -2.
\]