To analyze the given functions for Nashville and Houston:
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Houston's bike paths: \( y = 4(2)^x \) is an exponential function. This means the number of bike paths in Houston increases exponentially as \( x \) increases.
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Nashville's bike paths: \( y = 4x \) is a linear function. The number of bike paths in Nashville increases at a constant rate (specifically 4 paths for each unit increase in \( x \)).
Now, let's evaluate the statements:
A. False: The number of paths planned for Nashville is increasing linearly, not exponentially.
B. Cannot be determined from the information provided: We do not know which line represents Houston without seeing the graph.
C. Cannot be determined from the information provided: We do not know which line represents Nashville without seeing the graph.
D. True: As \( x \) increases, the exponential growth of Houston's paths will typically outpace the linear growth of Nashville’s paths, suggesting that at some point, the number of paths planned for Nashville will not exceed those in Houston. However, without knowing the specific values of \( x \) and the initial conditions, it is hard to say precisely when this occurs.
Based on this analysis, the most appropriate answer based on the information provided is:
D: The number of paths planned for Nashville will more than likely exceed the paths in Houston as the value of x increases.
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