Question

Which of the following includes the input-output pairs (2,4) and (3,8) ?

a. The arithmetic sequence an = 4n
b. The linear function f(n) = 2 + 4(n − 1)
c. The geometric sequence gn = 2ⁿ⁻¹
d. The exponential function h(n) = 2 . 2ⁿ⁻¹

Answer 1

The given exponential function h(n) = 2 · 2n-1 does include the input-output pairs (2,4) and (3,8). By substituting the input values into the function, the corresponding outputs match the ones given in the pairs, confirming their inclusion in the function.

Step-by-step explanation:

The question involves determining whether a given exponential function contains specific input-output pairs. Specifically, the pairs in question are (2,4) and (3,8). We are given the function h(n) = 2 · 2n-1. To verify if the pairs are included, we substitute the input values into the function and check if the output values match.

For the pair (2,4), substituting 2 for n gives us h(2) = 2 · 22-1 = 2 · 21 = 2 · 2 = 4, which matches the output of the pair. Similarly, for the pair (3,8), substituting 3 for n gives us h(3) = 2 · 23-1 = 2 · 22 = 2 · 4 = 8, which also matches the output of the pair.

Therefore, we can conclude that the given exponential function h(n) does indeed contain the pairs (2,4) and (3,8). This example also illustrates the concept of exponential growth, where each successive number in the function is multiplied by a consistent base number, in this case, 2. This is a fundamental property of exponential functions and can be seen in various real-life applications, such as population growth and compound interest.