To analyze the features of the function [tex]\( f(x) = 2 \cdot 3^x - 6 \)[/tex]:
1. Type of Function:
- The function [tex]\( f(x) \)[/tex] is an exponential function. This type of function typically takes the form [tex]\( a \cdot b^x + c \)[/tex].
2. Horizontal Asymptote:
- For an exponential function of the form [tex]\( f(x) = a \cdot b^x + c \)[/tex], the horizontal asymptote is [tex]\( y = c \)[/tex]. Here, the function has a horizontal asymptote of [tex]\( y = -6 \)[/tex].
3. Range of the Function:
- The range of an exponential function with a horizontal asymptote at [tex]\( y = -6 \)[/tex] and considering that the function grows indefinitely as [tex]\( x \)[/tex] increases, is [tex]\( (-6, \infty) \)[/tex].
4. Increasing or Decreasing:
- The function is increasing because the base of the exponential part, [tex]\( b = 3 \)[/tex], is greater than 1.
5. Domain of the Function:
- The domain of an exponential function is all real numbers. Therefore, the domain of this function is [tex]\( (-\infty, \infty) \)[/tex].
6. End Behavior:
- As [tex]\( x \to -\infty \)[/tex], the value of [tex]\( f(x) \)[/tex] approaches the horizontal asymptote. Hence, [tex]\( f(x) \to -6 \)[/tex].
- As [tex]\( x \to \infty \)[/tex], the exponential term grows infinitely large. Therefore, [tex]\( f(x) \to \infty \)[/tex].
Putting it all together, we have:
The function [tex]\( f(x) \)[/tex] is an exponential function with a horizontal asymptote of [tex]\( y = -6 \)[/tex]. The range of the function is [tex]\( (-6, \infty) \)[/tex], and it is increasing on its domain of [tex]\( (-\infty, \infty) \)[/tex]. The end behavior on the left side is as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -6 \)[/tex], and the end behavior on the right side is as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
