Determine the range values from an equation.
Consider the functions [tex]f(x)=\left(\frac{4}{5}\right)^x[/tex] and [tex]g(x)=\left(\frac{4}{5}\right)^x+6[/tex]. What are the ranges of the two functions?
[tex]\begin{array}{l}
f(x):\{y \mid y\ \textgreater \ [/tex]
[tex]g(x):\{y \mid y\ \textgreater \ [/tex]
Answer (2)
The range of f ( x ) = ( 5 4 ) x is determined by the behavior of the exponential function with a base between 0 and 1, resulting in 0"> y > 0 .
The range of g ( x ) = ( 5 4 ) x + 6 is found by shifting the range of f ( x ) vertically upward by 6 units, resulting in 6"> y > 6 .
Therefore, the range of f ( x ) is all positive real numbers, and the range of g ( x ) is all real numbers greater than 6.
The final answer is: 0\}"> f ( x ) : { y ∣ y > 0 } and 6\}"> g ( x ) : { y ∣ y > 6 } .
Explanation
Understanding the Problem We are given two functions, f ( x ) = ( 5 4 ) x and g ( x ) = ( 5 4 ) x + 6 , and we need to determine their ranges. The range of a function is the set of all possible output values (y-values) that the function can produce.
Finding the Range of f(x) For the function f ( x ) = ( 5 4 ) x , we observe that the base 5 4 is between 0 and 1. This means that as x increases, the function value decreases, and as x decreases, the function value increases. Since any number between 0 and 1 raised to any power will always be greater than 0, f ( x ) will always be greater than 0. As x approaches infinity, f ( x ) approaches 0, but never actually reaches 0. As x approaches negative infinity, f ( x ) approaches infinity. Therefore, the range of f ( x ) is all positive real numbers, or 0"> y > 0 .
Finding the Range of g(x) Now, let's consider the function g ( x ) = ( 5 4 ) x + 6 . This function is simply the function f ( x ) shifted vertically upward by 6 units. This means that every output value of f ( x ) is increased by 6 to obtain the output value of g ( x ) . Since the range of f ( x ) is 0"> y > 0 , the range of g ( x ) will be 0 + 6"> y > 0 + 6 , which simplifies to 6"> y > 6 .
Final Answer Therefore, the ranges of the two functions are:
0\}"> f ( x ) : { y ∣ y > 0 }
6\}"> g ( x ) : { y ∣ y > 6 }
Examples
Understanding the range of exponential functions is crucial in various real-world applications. For instance, in finance, when modeling the depreciation of an asset over time, an exponential function can be used. The range of this function would tell you the possible values the asset's worth can take, helping in financial planning and risk assessment. Similarly, in biology, when studying population decay or growth, the range of the exponential function provides insights into the possible population sizes.
The range of the function f ( x ) = ( 5 4 ) x is 0"> y > 0 , meaning it includes all positive real numbers. The range of the function g ( x ) = ( 5 4 ) x + 6 is 6"> y > 6 , indicating it includes all real numbers greater than 6.
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