Understanding Inverse Functions

The video tutorial explains how to find the inverse of a function, specifically a rational function. It begins with an introduction to the function f(x) = (2x + 5) / (4 - 3x) and the concept of an inverse function. The tutorial then covers the domain and range of functions and how an inverse function works. The main focus is on algebraic manipulation to find the inverse, swapping x and y, and solving for y. The video concludes with the final expression of the inverse function and an alternative way to express it.

What is the initial function f(x) given in the video?

• f(x) = 2x + 5 / (4 - 3x)

• f(x) = 4x + 3 / (5 - 2x)

• f(x) = 3x + 4 / (2 - 5x)

• f(x) = 5x + 2 / (3 - 4x)

What does the inverse of a function do?

• It adds a constant to the function.

• It squares the function's input.

• It doubles the function's output.

• It swaps the domain and range.

In the context of functions, what is the domain?

• The set of all positive values.

• The set of all possible output values.

• The set of all negative values.

• The set of all possible input values.

What is the first step in finding the inverse of the function?

• Multiply both sides by 2.

• Divide both sides by 3.

• Add a constant to both sides.

• Swap x and y in the equation.

What operation is performed after swapping x and y to solve for y?

• Add 5 to both sides.

• Divide by 2.

• Multiply both sides by the denominator.

• Subtract 4 from both sides.

What is the purpose of factoring out y in the equation?

• To isolate y on one side of the equation.

• To eliminate x from the equation.

• To simplify the numerator.

• To change the sign of the equation.

What is the final expression for the inverse function?

• y = (4x - 5) / (3x + 2)

• y = (4x + 5) / (3x - 2)

• y = (5 - 4x) / (3x + 2)

• y = (5 + 4x) / (3x - 2)

How can the expression for the inverse function be simplified?

• By multiplying the numerator and denominator by -1.

• By adding 1 to both the numerator and denominator.

• By dividing the numerator by 2.

• By subtracting 3 from the denominator.

What is the result of multiplying the numerator and denominator by -1?

• The equation remains unchanged.

• The equation becomes undefined.

• The equation becomes zero.

• The signs of both the numerator and denominator are reversed.

What is the significance of finding the inverse of a function?

• It squares the input of the original function.

• It doubles the output of the original function.

• It adds a constant to the original function.

• It allows us to reverse the effect of the original function.