$\begin{array}{l}
f(x)=\left(\frac{2}{5}\right)^x \\
g(x)=\left(\frac{2}{5}\right)^x-3
\end{array}$
Which statement about $f(x)$ and its translation, $g(x)$, is true?
A. The range of $g(x)$ is, ${y \mid y>0}$ and the range of $f(x)$ is ${y \mid y>-3}$.
B. The range of $g(x)$ is, ${y \mid y>3}$ and the range of $f(x)$ is ${y \mid y>0}$.
C. The asymptote of $g(x)$ is the asymptote of $f(x)$ shifted three units down.
D. The asymptote of $g(x)$ is the asymptote of $f(x)$ shifted three units up.
Answer (1)
The range of f ( x ) = ( 5 2 ) x is 0"> y > 0 .
The range of g ( x ) = ( 5 2 ) x − 3 is -3"> y > − 3 .
The horizontal asymptote of f ( x ) is y = 0 .
The horizontal asymptote of g ( x ) is y = − 3 , which is the asymptote of f ( x ) shifted three units down. Therefore, the correct answer is: The asymptote of g ( x ) is the asymptote of f ( x ) shifted three units down.
Explanation
Understanding the Problem We are given two functions: f ( x ) = ( 5 2 ) x and g ( x ) = ( 5 2 ) x − 3 . We need to determine which statement about f ( x ) and its translation, g ( x ) , is true.
Finding the Range of f(x) Let's analyze the range of f ( x ) . Since ( 5 2 ) x is always positive for any real number x , the range of f ( x ) is all positive real numbers, i.e., 0"> y > 0 . In set notation, this is 0}"> y ∣ y > 0 .
Finding the Range of g(x) Now let's analyze the range of g ( x ) . The function g ( x ) is a vertical translation of f ( x ) by 3 units down. Therefore, the range of g ( x ) is the range of f ( x ) shifted 3 units down. So, the range of g ( x ) is -3"> y > − 3 . In set notation, this is -3}"> y ∣ y > − 3 .
Finding the Asymptote of f(x) Next, let's find the horizontal asymptote of f ( x ) . As x approaches infinity, ( 5 2 ) x approaches 0. Therefore, the horizontal asymptote of f ( x ) is y = 0 .
Finding the Asymptote of g(x) Now let's find the horizontal asymptote of g ( x ) . Since g ( x ) = f ( x ) − 3 , the horizontal asymptote of g ( x ) is the horizontal asymptote of f ( x ) shifted 3 units down. So, the horizontal asymptote of g ( x ) is y = − 3 . This means the asymptote of g ( x ) is the asymptote of f ( x ) shifted three units down.
Conclusion Comparing our findings with the given statements, we see that the statement "The asymptote of g ( x ) is the asymptote of f ( x ) shifted three units down" is true.
Examples
Exponential functions and their translations are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. Understanding how vertical translations affect the range and asymptotes of these functions is crucial for making accurate predictions and informed decisions. 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, then g ( x ) = f ( x ) − b could represent the same decay process but with a background radiation level of b . Knowing the asymptote of g ( x ) helps us determine the minimum level of radiation that will always be present.
