Finding a piecewise function definition from graph | Algebra II | Khan Academy

TL;DR

The graph represents a piecewise continuous function with two intervals: one defined by the square root of x plus 2, and the other by x minus 4 to the third power.

Transcript

Select the piecewise function whose graph is shown below. Or I guess we should say to the right. I copied and pasted it so it's on the right now. So we have this piecewise continuous function. So it's not defined for x being negative 2 or lower. But then starting at x greater than negative 2, it starts being defined. It's continuous all the way unt... Read More

Key Insights

• 📈 The graph represents a piecewise continuous function with two defined intervals.
• ☺️ The function is defined by the square root of x plus 2 from x greater than -2 but less than or equal to 2.
• ☺️ The function is defined by x minus 4 to the third power from x greater than 2 but less than or equal to 6.
• ☺️ The function has a discontinuity at x equals 2.
• ☺️ The function is undefined for x less than or equal to -2 and x greater than 6.
• 🛀 The graph shows a shift and transformation of the square root and cubic functions.
• 👈 The function has specific values and characteristics at certain points, such as x equals -2, -1, 2, 4, and 6.

Questions & Answers

Q: What are the intervals where the function is defined?

The function is defined for x greater than -2 but less than or equal to 2, and from x greater than 2 but less than or equal to 6.

Q: What is the formula for the function in the first interval?

In the first interval, the function is represented by the square root of x plus 2, where x is greater than -2 and less than or equal to 2.

Q: What is the formula for the function in the second interval?

In the second interval, the function is represented by x minus 4 to the third power, where x is greater than 2 and less than or equal to 6.

Q: Why is the function discontinuous at x equals 2?

The function is discontinuous at x equals 2 because there is a jump from the first interval to the second interval in the graph.

Summary & Key Takeaways

• The graph shows a piecewise continuous function with a defined interval for x greater than -2, another interval from -2 to 2, and a final interval for x less than or equal to 6.

• The function consists of two parts: the square root of x plus 2 for x greater than -2 but less than or equal to 2, and x minus 4 to the third power for x greater than 2 but less than or equal to 6.

• The function is continuous except for a discontinuity at x equals 2.