What is the range of the function $f(x)=-2\left(6^x\right)+3$?
A. $(-\infty,-2]$
B. $(-\infty, 3)$
C. $[-2, \infty)$
D. $[3, \infty)$
Answer (2)
Analyze the exponential function 6 x , which has a range of ( 0 , ∞ ) .
Multiply by -2 to get − 2 ( 6 x ) , which has a range of ( − ∞ , 0 ) .
Add 3 to get f ( x ) = − 2 ( 6 x ) + 3 , which has a range of ( − ∞ , 3 ) .
The range of the function f ( x ) is ( − ∞ , 3 ) . ( − ∞ , 3 )
Explanation
Understanding the Problem We are given the function f ( x ) = − 2 ( 6 x ) + 3 and asked to find its range. The range is the set of all possible output values of the function.
Analyzing the Exponential Part First, let's analyze the exponential part, 6 x . Since the base 6 is greater than 1, the exponential function 6 x is always positive. As x approaches negative infinity, 6 x approaches 0. As x approaches positive infinity, 6 x approaches infinity. Therefore, the range of 6 x is ( 0 , ∞ ) .
Considering the Term -2(6^x) Next, we consider the term − 2 ( 6 x ) . Multiplying by -2 flips the interval and scales it. So, as 6 x ranges from ( 0 , ∞ ) , − 2 ( 6 x ) ranges from ( − ∞ , 0 ) .
Finding the Range of f(x) Finally, we add 3 to the term − 2 ( 6 x ) to obtain f ( x ) = − 2 ( 6 x ) + 3 . Adding 3 shifts the interval, so the range of − 2 ( 6 x ) + 3 is ( − ∞ , 3 ) .
Final Answer Therefore, the range of the function f ( x ) = − 2 ( 6 x ) + 3 is ( − ∞ , 3 ) .
Examples
Consider a scenario where the temperature of an object decreases exponentially over time, but it's also influenced by a constant external factor. The function f ( x ) = − 2 ( 6 x ) + 3 could model this, where x is time, 6 x represents the exponential decay, -2 is a scaling factor, and +3 is the constant external temperature. Understanding the range of this function helps predict the possible temperature values the object can reach, which is crucial in various applications like material science or climate modeling.
The range of the function f ( x ) = − 2 ( 6 x ) + 3 is determined by the behavior of the exponential function and its transformations. After analyzing it step-by-step, we find that the range is ( − ∞ , 3 ) . Therefore, the correct choice is option B: ( − ∞ , 3 ) .
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