Understanding Functions and Their Inverses

The video tutorial explains how to work with the function f(x) = 1/4 x^2 - 8 over a closed interval. It covers calculating f(4) both graphically and algebraically, finding the derivative to determine the slope of the tangent line, graphing the inverse function, and calculating the derivative of the inverse function using reciprocal relationships. The tutorial emphasizes understanding the graphical and algebraic relationships between a function and its inverse.

What is the value of f(4) for the function f(x) = 1/4 x^2 - 8?

• -4

• 4

• 0

• 8

How can the value of f(4) be determined?

• Neither method

• Both graphically and algebraically

• Only algebraically

• Only graphically

What is the derivative of f(x) = 1/4 x^2 - 8?

• 1/2 x

• 1/4 x

• 1/4 x - 8

• 1/2 x - 8

What is the slope of the tangent line to f(x) at x = 4?

• 1

• 4

• 2

• 3

How are the graphs of a function and its inverse related?

• They are parallel

• They are perpendicular

• They are reflections across the line y = x

• They are identical

Which point is on the graph of f inverse if (4, -4) is on f(x)?

• (-4, 4)

• (4, 4)

• (4, -4)

• (-4, -4)

What is the relationship between the slopes of tangent lines of a function and its inverse?

• They are unrelated

• They are negative reciprocals

• They are reciprocals

• They are equal

How can the derivative of the inverse function at a point be found?

• By finding the inverse function and differentiating

• By using the reciprocal of the derivative of the original function

• By graphing the inverse function

• By using the original function's value

What is the derivative of the inverse function at x = -4?

• 0

• 1

• 2

• 1/2

If f(4) = -4, what is f inverse(-4)?

• 8

• 4

• -4

• 0