Math 30 - Day 8 - Inverse Functions

Overview

Christopher Lee explains inverse functions as operations that reverse a function’s input and output, then shows how to verify inverses with composition and find them by solving for one variable and swapping x and y. Examples include converting Celsius back to Fahrenheit, checking polynomial and rational-function pairs, and handling one-to-one requirements by restricting domains and swapping the original function’s domain and range.

Key takeaways

An inverse reverses input-output pairs: if f(a) = b, then f⁻¹(b) = a, so every ordered pair (a, b) becomes (b, a).
To verify that f and g are inverses, compose them and check that the result is x; for example, 1/(x + 2) and 1/x − 2 undo each other.
The notation f⁻¹(x) means inverse function, not reciprocal; the reciprocal is 1/f(x).
A function must be one-to-one to have an inverse function on its full domain; x² fails because both 3 and −3 map to 9, but restricting to x ≥ 0 gives inverse √x.
For inverse functions, the original range becomes the inverse domain and the original domain becomes the inverse range; for f(x) = 2 − √x, this makes the inverse domain (−∞, 2].
Solving C = 5/9(F − 32) for F gives the practical inverse conversion F = 9/5C + 32.

Chapters

0:00 Reversible Heat Pumps Introduce the Idea of Running a Function Backward

Christopher Lee opens with a reversible heat pump, which can move heat out of a building for cooling or into it for heating.
The two operating directions motivate the lesson’s central question: whether a function machine can also run backward.

1:00 Celsius-to-Fahrenheit Conversion Creates a Need for an Inverse Formula

The Celsius conversion formula is C = 5/9(F − 32); substituting 75°F gives approximately 24°C.
Betty’s Milan forecast includes Celsius readings from 18°C to 30°C, and repeatedly solving the formula for Fahrenheit would be tedious.

6:00 Why a Direct Fahrenheit Formula Is More Convenient

A formula that accepts Celsius as input and returns Fahrenheit as output avoids solving the same equation for each forecast reading.
An inverse function reverses the original function’s mapping: its input is the original output, and its output is the original input.

8:30 Inverse Notation and the Function-Composition Identity

The notation f⁻¹(x) names the inverse function; it does not mean 1/f(x), which is the reciprocal.
When f and its inverse are composed, the original input is recovered: f⁻¹(f(x)) = x, and likewise f(f⁻¹(x)) = x on their domains.

12:00 Composition Tests Whether Two Functions Undo Each Other

To verify a proposed inverse g of f, substitute one function into the other and check whether the result simplifies to x.
For example, with f(x) = 4x and g(x) = x/4, composition returns x, illustrating how an inverse undoes the original operation.

15:00 Inverse Coordinate Pairs Swap Their Inputs and Outputs

If f(2) = 4, then f⁻¹(4) = 2; if f(5) = 12, then f⁻¹(12) = 5.
In a table or ordered pair, the inverse swaps each (x, y) to (y, x), such as (2, 4) becoming (4, 2).

19:00 A Practical Procedure for Testing Inverse Relationships

For functions f and g, calculate either f(g(x)) or g(f(x)) and check whether it equals x.
Lee says one composition is sufficient for the class exercises: if it does not simplify to x, the functions are not inverses.

22:00 Verify the Rational Inverse Pair 1/(x + 2) and 1/x − 2

The example uses f(x) = 1/(x + 2) and g(x) = 1/x − 2.
Substituting g into f cancels the −2 and +2, leaving 1/(1/x) = x; the reverse composition also simplifies to x.

23:00 Inverse Operations Unwind in Reverse Order

The inverse must undo the original function’s operations in reverse order, such as undoing a cube with a cube root.
For f(x) = x³ − 4, g(x) = ∛(x + 4) reverses the subtraction and cubing.

29:00 Composition Confirms a Cube-and-Cube-Root Inverse Pair

For f(x) = x³ − 4 and g(x) = ∛(x + 4), f(g(x)) simplifies to x because the cube and cube root cancel and the constants offset.
The reverse composition also returns x; Lee emphasizes simplifying the additions and subtractions before applying the outer power.

41:00 A Scalar Multiple of x³ Is Not the Inverse of x³

The example tests f(x) = x³ against g(x) = x/3.
The compositions produce x³/27 and x³/3 rather than x, so the proposed functions are not inverses.

49:00 A Shifted Cube and Cube Root Form Another Inverse Pair

For f(x) = (x − 1)³ and g(x) = ∛x + 1, substituting either function into the other cancels the shift and the cube-root/cube operations.
Lee cautions students to simplify the inside expression correctly before applying an outside exponent.

54:00 Why x² Needs a Restricted Domain to Have an Inverse Function

The output 9 of f(x) = x² comes from both inputs 3 and −3, so reversing the mapping would assign two outputs to one input.
The parabola fails the horizontal line test; restricting x² to x ≥ 0 makes it one-to-one and gives the inverse √x.

1:00:00 Inverse Functions Swap Domain and Range

The range of f becomes the domain of f⁻¹, while the domain of f becomes the range of f⁻¹.
If f(x) = (x − 1)² is restricted to x ≥ 1, its inverse is √x + 1, with the restricted domain becoming the inverse’s range.

1:02:00 Find an Inverse from a Formula by Solving for x

Lee’s procedure is to confirm the function is one-to-one, solve the equation for x in terms of y, then swap variable roles.
The final expression is written as the inverse function, often replacing y with f⁻¹(x).

1:03:00 Derive the Fahrenheit Formula from the Celsius Equation

Starting with C = 5/9(F − 32), multiply both sides by 9/5 and then add 32.
The inverse conversion is F = 9/5C + 32, so a Celsius forecast value can be substituted directly to obtain Fahrenheit.

1:07:00 Solve a Linear Formula for Its Inverse

For y = 1/3x − 5, multiplying by 3 and adding 5 gives x = 3y + 5.
Swapping the variables yields the inverse rule y = 3x + 5; the algebra keeps both sides of the equation balanced.

1:17:00 Find the Inverse of a Shifted Reciprocal Function

The example is f(x) = 2/(x − 3) + 4; subtracting 4 isolates the reciprocal expression.
Rearranging for x and swapping variables gives f⁻¹(x) = 2/(x − 4) + 3, with excluded values carried through the domain and range.

1:22:00 Use Reversed Algebraic Steps to Invert a Radical Function

For f(x) = 2 + √(x − 4), subtract 2 and square both sides to isolate x.
After swapping variables, the inverse is f⁻¹(x) = (x − 2)² + 4; because the original square-root function has range x ≥ 2, that becomes the inverse’s domain.

1:28:00 Restrict the Inverse Domain to Preserve One-to-One Behavior

A quadratic expression can have a real-valued formula for every real input yet still fail to be one-to-one across its full domain.
For the inverse of 2 + √(x − 4), the domain is restricted to x ≥ 2 so that the inverse matches the original function’s range.

1:35:00 Invert 2 − √x and Identify Functions That Are Their Own Inverses

For f(x) = 2 − √x, the original domain is [0, ∞) and its range is (−∞, 2]; the inverse is f⁻¹(x) = (2 − x)² with domain (−∞, 2].
The identity function and reciprocal function are examples of self-inverse functions; Lee also notes that f(x) = c − x is its own inverse.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Christopher Lee.