The Weird Graph of y = x^x | Summary and Q&A

TL;DR

The graph of y = x^x is continuous on the right side and discontinuous on the left side due to the presence of imaginary numbers and positive results from fractional exponents.

Key Insights

• 📈 The graph of y = x^x is created using an online graphing calculator like Desmos.com.
• 🗯️ The graph is continuous on the right side when positive values are plugged in.
• 🔌 Plugging in negative values can result in imaginary numbers due to the presence of fractional exponents.
• ❎ Some negative values can still give real outputs, depending on their fractional exponents.
• 🗨️ Positive values on the left side occur when fractional exponents make negative numbers positive.
• 📈 The behavior of the graph is determined by the nature of fractional exponents.
• 🗨️ Impossibility of negative outputs with certain values results in the discontinuity on the left side.

Transcript

in this video we're going to talk about the weird graph of y equals x raised to the x so this is an online graphing calculator at desmos.com and you can graph any function with this website you can graph the linear function y equals x y equals x squared trigonometric functions like sine x or tangent x but what we want to focus on is this one x to t... Read More

Questions & Answers

Q: Why are some values positive on the left side of the y-axis and some negative?

Some values on the left side are positive because certain fractional exponents, like 2/5, can turn negative values into positive ones by making the numerator even.

Q: Why do we see values below the x-axis and others above it?

Values below the x-axis occur when negative numbers are plugged in, resulting in negative outputs. Values above the x-axis can arise from fractional exponents that make negative values positive.

Q: Why is the graph continuous on the right side but not on the left side?

The graph is continuous on the right side because any positive value for x will produce a valid output. The left side is discontinuous due to the presence of imaginary numbers when negative values are plugged in.

Q: Why can't all negative values be plugged in, resulting in imaginary numbers?

Negative values can only be plugged in if the fractional exponents convert them into odd roots, yielding real negative outputs. But some fractional exponents will result in even roots, which give imaginary numbers.

Summary & Key Takeaways

• The graph of y = x^x can be created using an online graphing calculator like Desmos.com.

• The graph is continuous on the right side when positive values are plugged in, but discontinuous on the left side when negative values are plugged in.

• Plugging in negative values can result in imaginary numbers, while plugging in certain fractional exponents can yield negative or positive results.