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]
[tex]\end{array}
[/tex]
Answer (1)
The function f ( x ) = ( 5 4 ) x is a decreasing exponential function, and its range is all positive real numbers.
The range of f ( x ) is 0"> y > 0 .
The function g ( x ) = ( 5 4 ) x + 6 is a vertical shift of f ( x ) by 6 units.
The range of g ( x ) is 6"> y > 6 , so the final answer is 0 , 6 .
Explanation
Understanding the Functions We are given two exponential functions: f ( x ) = ( 5 4 ) x and g ( x ) = ( 5 4 ) x + 6 . We need to determine the range of each function.
Analyzing the Range of f(x) For f ( x ) = ( 5 4 ) x , since the base 5 4 is between 0 and 1, the function is a decreasing exponential function. 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, i.e., 0"> y > 0 .
Analyzing the Range of g(x) For g ( x ) = ( 5 4 ) x + 6 , this is the function f ( x ) shifted vertically upwards by 6 units. Since the range of f ( x ) is 0"> y > 0 , the range of g ( x ) will be 0 + 6"> y > 0 + 6 , which means 6"> y > 6 .
Final Ranges Thus, the range of f ( x ) is 0"> y > 0 and the range of g ( x ) is 6"> y > 6 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding the range of an exponential function helps in predicting the possible values of these phenomena. For example, if we model the decay of a radioactive substance with f ( x ) = a ⋅ ( 2 1 ) x , where a is the initial amount and x is time, knowing the range ( 0"> y > 0 ) tells us that the amount of the substance will always be positive, approaching zero but never reaching it.
