Using the horizontal line test, determine whether the function is one-to-one.
[tex]f(x)=2.9^x[/tex]
Is the function one-to-one?
A. No
B. Yes
Answer (2)
The horizontal line test determines if a function is one-to-one.
f ( x ) = 2. 9 x is an exponential function with a base greater than 1, making it strictly increasing.
Strictly increasing functions pass the horizontal line test.
Therefore, the function is one-to-one: Yes .
Explanation
Understanding the Problem The problem asks us to determine whether the function f ( x ) = 2. 9 x is one-to-one using the horizontal line test.
Understanding the Horizontal Line Test The horizontal line test states that a function is one-to-one if and only if every horizontal line intersects the graph of the function at most once. In simpler terms, if any horizontal line drawn on the graph intersects the function more than once, the function is not one-to-one.
Analyzing the Function The function f ( x ) = 2. 9 x is an exponential function. Exponential functions of the form f ( x ) = a x , where 0"> a > 0 and a = 1 , are either strictly increasing (if 1"> a > 1 ) or strictly decreasing (if 0 < a < 1 ). In our case, a = 2.9 , which is greater than 1, so the function is strictly increasing.
Applying the Horizontal Line Test A strictly increasing function means that as x increases, f ( x ) also increases. This implies that no horizontal line can intersect the graph of f ( x ) more than once. If a horizontal line were to intersect the graph twice, it would mean that there are two different x values that produce the same f ( x ) value, which contradicts the definition of a strictly increasing function.
Conclusion Since f ( x ) = 2. 9 x is a strictly increasing function, it passes the horizontal line test. Therefore, the function is one-to-one.
Examples
One-to-one functions are essential in cryptography, where each input must map to a unique output to ensure secure encryption. For example, the exponential function can model population growth, where each time period corresponds to a unique population size. Determining if a function is one-to-one helps ensure that each input has a unique output, which is crucial for secure data transmission and accurate modeling of real-world phenomena.
The function f ( x ) = 2. 9 x is one-to-one as it is a strictly increasing exponential function. It passes the horizontal line test, meaning every horizontal line intersects the graph at most once. Therefore, the answer is: Yes .
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