x & f(x) 0 & 100 1 & 120 2 & 144 3 & 172.8

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Question

x & f(x) 0 & 100 1 & 120 2 & 144 3 & 172.8

Answer 1

The function is \(f(x) = 100 \times 1.2^x\).

Answer 2

## Step 1: Observing the given values We are given a table with two columns, where the left column represents values of \(x\) and the right column represents corresponding values of a function \(f(x)\). - When \(x = 0\), \(f(x) = 100\). - When \(x = 1\), \(f(x) = 120\). - When \(x = 2\), \(f(x) = 144\). - When \(x = 3\), \(f(x) = 172.8\). ## Step 2: Identifying the pattern - The value of \(f(x)\) increases as \(x\) increases. - The ratio of consecutive \(f(x)\) values seems consistent. To confirm this, we calculate the ratio of each pair of consecutive values: - \(\frac{120}{100} = 1.2\) - \(\frac{144}{120} = 1.2\) - \(\frac{172.8}{144} = 1.2\) - This consistent ratio suggests that the function \(f(x)\) could be an exponential function where each step involves multiplying by a constant factor, in this case, 1.2. ## Step 3: Formulating the function - Given the pattern, the function can be expressed as \(f(x) = 100 \times 1.2^x\). This formula fits all the given data points. ## Step 4: Verifying the function - For \(x = 0\), \(f(0) = 100 \times 1.2^0 = 100\). - For \(x = 1\), \(f(1) = 100 \times 1.2^1 = 120\). - For \(x = 2\), \(f(2) = 100 \times 1.2^2 = 144\). - For \(x = 3\), \(f(3) = 100 \times 1.2^3 = 172.8\). - The function \(f(x) = 100 \times 1.2^x\) correctly produces all the values in the table. ## Step 5: Conclusion - The function \(f(x)\) is an exponential function where the base value is 100, and it is multiplied by 1.2 for each increase in \(x\) by 1.

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x & f(x) 0 & 100 1 & 120 2 & 144 3 & 172.8

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