Inverse of One-to-One Function | Grade 11- General Mathematics

Summary

This educational video tutorial teaches viewers how to find the inverse of one-to-one functions. It covers three examples: solving for the inverse of a linear function (f(x) = 5x - 8), a rational function (f(x) = (x - 4)/(2x - 7)), and a cubic function (f(x) = 2x^3 - 5). The process involves changing f(x) to y, swapping x and y, solving for y, and simplifying. The video concludes with the inverse functions for each example, encouraging viewers to like and subscribe for more educational content.

Takeaways

• 📕 To find the inverse of a one-to-one function, you first replace f(x) with y.
• 📝 Interchange the variables x and y to reflect the inverse relationship.
• 💵 Solve for y to derive the inverse function.
• 💲 For a linear function like f(x) = 5x - 8, isolate y by transposing terms and dividing by the coefficient.
• 💳 The inverse of f(x) = 5x - 8 is y = (x + 8)/5.
• 📘 For rational expressions, cross-multiply to eliminate the fraction.
• 💴 Group terms with y on one side and terms without y on the other side to isolate y.
• 💹 The inverse of f(x) = (x - 4)/(2x - 7) is y = (7x - 4)/(2x - 1).
• 📗 For functions involving exponents, such as f(x) = 2x^3 - 5, solve for y by transposing and dividing to isolate the cube root.
• 💱 The inverse of f(x) = 2x^3 - 5 is y = √x + 5/2∛.
• 📝 The video provides a step-by-step guide on how to find the inverse of different types of functions, including linear, rational, and cubic functions.

Q & A

What is the first step in finding the inverse of a one-to-one function?

The first step is to change f(x) to y, so that y equals the function expression.

How do you interchange variables to find the inverse function?

You interchange x and y variables in the equation to express x in terms of y.

What is the inverse function of f(x) = 5x - 8?

The inverse function of f(x) = 5x - 8 is x + 8/5.

How do you solve for y in the equation x = 5y - 8 after interchanging variables?

You transpose -8 to the left side to get +8 and then divide both sides by 5 to isolate y.

What is the process for solving the inverse of a rational function like f(x) = (x - 4)/(2x - 7)?

You interchange x and y, cross multiply to eliminate the fraction, and then group terms with y on one side and terms without y on the other side.

How do you handle the rational expression after interchanging x and y in f(x) = (x - 4)/(2x - 7)?

You cross multiply to get rid of the fraction, then rearrange terms to isolate y.

What is the inverse function of f(x) = (x - 4)/(2x - 7)?

The inverse function of f(x) = (x - 4)/(2x - 7) is 7x - 4/(2x - 1).

How do you solve for y in the equation x = y^3 - 5 after interchanging variables?

You transpose -5 to the other side to get +5, then divide both sides by 2, and finally take the cube root of both sides.

What is the inverse function of f(x) = 2x^3 - 5?

The inverse function of f(x) = 2x^3 - 5 is the cube root of (x + 5)/2.

Why is it necessary to cube root both sides after isolating y in the equation x = y^3 - 5?

You cube root both sides to eliminate the power of 3 on the right side and isolate y.

What does it mean when the inverse function is represented with a negative exponent on y?

A negative exponent on y indicates that the function is the inverse of the original function.