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.) $$ x=1.1 $$

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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 $$ x = 1.1 $$.
At $$ x = 1.1 $$, the graph of $$ g(x) = e^{x} $$ surpasses the graph of $$ f(x) = x^{10} $$.

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.) \( x=1.1 \)

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