Solution:
Given the exponential function \( f(x) = 3 \cdot 2^x - 1 \).
1. Understanding the Function:
The given function is an exponential function with the form \( f(x) = a \cdot b^x + c \), where:
- \( a = 3 \) (scaling factor)
- \( b = 2 \) (base, greater than 1, leading to exponential growth)
- \( c = -1 \) (vertical shift)
2. Finding the Domain:
In an exponential function, the exponent \( x \) can take any real value because the base \( b \) will always yield a valid result.
Thus, the domain is all real numbers:
\[ \text{Domain} = (-\infty, \infty) \]
3. Finding the Range:
Since the base of the exponential function \( b = 2 \) is positive and greater than 1, the function will grow exponentially as \( x \) increases.
The coefficient \( a = 3 \) scales the output, and the constant \( c = -1 \) shifts the entire function downward by 1.
Since exponential functions with positive bases never produce zero or negative results, the minimum value of \( 3 \cdot 2^x \) is 0. This result is shifted down by 1, meaning the minimum output is:
\[ \text{Minimum Value} = -1 \]
As \( x \) increases, \( f(x) \) continues to grow towards positive infinity.
Therefore, the range of the function is:
\[ \text{Range} = (-1, \infty) \]