A graphing calculator is recommended. Compare the functions \( f(x)=x^{10} \) and \( g(x)=e^{x} \) by graphing both functions in several viewing rectangles. When does the graph of \( g \) finally surpass the graph of \( f ? \) (Round your answer to one decimal place.) \( x=1.1 \)

To compare the functions \( f(x) = x^{10} \) and \( g(x) = e^{x} \) by graphing both functions in several viewing rectangles, we can use a graphing calculator. To find when the graph of \( g \) finally surpasses the graph of \( f \), we need to find the point where \( g(x) > f(x) \). Given that \( x = 1.1 \), we can substitute this value into both functions to determine when the graph of \( g \) surpasses the graph of \( f \). Let's calculate the values of \( f(1.1) \) and \( g(1.1) \) to determine when the graph of \( g \) surpasses the graph of \( f \). Evaluate the expression by following steps: - step0: Evaluate: \(x^{10}\) - step1: Substitute: \(1.1^{10}\) - step2: Convert the expressions: \(\left(\frac{11}{10}\right)^{10}\) - step3: Evaluate the power: \(\frac{11^{10}}{10^{10}}\) Substitute \( x = 1.1 \) into the function \( f(x) = x^{10} \) to find \( f(1.1) \): \[ f(1.1) = \frac{11^{10}}{10^{10}} \approx 2.593742 \] Now, let's calculate the value of \( g(1.1) \) by substituting \( x = 1.1 \) into the function \( g(x) = e^{x} \). Evaluate the expression by following steps: - step0: Evaluate: \(e^{x}\) - step1: Substitute: \(e^{1.1}\) - step2: Convert the expressions: \(e^{\frac{11}{10}}\) - step3: Transform the expression: \(\sqrt[10]{e^{11}}\) - step4: Rewrite the exponent: \(\sqrt[10]{e^{10+1}}\) - step5: Expand the expression: \(\sqrt[10]{e^{10}\times e}\) - step6: Use the properties of radicals: \(\sqrt[10]{e^{10}}\times \sqrt[10]{e}\) - step7: Simplify the root: \(e\sqrt[10]{e}\) Substitute \( x = 1.1 \) into the function \( g(x) = e^{x} \) to find \( g(1.1) \): \[ g(1.1) \approx 3.004166 \] Now, we can compare the values of \( f(1.1) \) and \( g(1.1) \) to determine when the graph of \( g \) surpasses the graph of \( f \) at \( x = 1.1 \).