Verifying Inverse Functions | Precalculus

The Organic Chemistry Tutor
11 Feb 201812:06
EducationalLearning
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TLDRThis video tutorial explores the concept of inverse functions, demonstrating how to determine if two functions are inverses by showing that their composition results in the identity function. It walks through several examples, including squaring and square root operations, and provides step-by-step solutions to verify if functions are inverses by composition and by finding the explicit inverse function. The video also includes a practice problem involving complex fractions and concludes with examples that illustrate both inverse and non-inverse relationships between functions.

Takeaways
  • 🔍 To determine if two functions are inverses, one must show that the composition of the functions, f(g(x)) and g(f(x)), both equal x.
  • 📚 The first example demonstrates that the functions f(x) = x^2 + 5 and g(x) = √(x - 5) are inverses because f(g(x)) and g(f(x)) simplify to x.
  • 📝 The second example with f(x) = 3x + 8 and g(x) = 8x^2 - 3 shows that they are not inverses as f(g(x)) does not simplify to x.
  • 🔄 The third example involves f(x) = 7x + 5 and g(x) = (x - 5) / 7, and it is shown that they are inverses since both compositions equal x.
  • 📉 In the fourth example, f(x) = 5/x + 3 and g(x) = 5 / (x - 3) are proven to be inverses through simplification of their compositions.
  • 📌 The fifth example with f(x) = 8x + 7 and g(x) = 7/x - 8 does not result in inverse functions as f(g(x)) does not equal x.
  • 🧩 The sixth and final example confirms that f(x) = x^3 + 5 and g(x) = ∛(x - 5) are inverses, as their compositions both result in x.
  • 📑 The script provides a step-by-step method to verify if functions are inverses by composing them and simplifying the result.
  • 📐 An alternative method to confirm inverse functions is by finding the inverse function of f and comparing it to g.
  • 📈 The script uses algebraic manipulation to simplify compositions and determine the relationship between functions.
  • 📘 The examples given in the script are practical applications of the concept of inverse functions in mathematics.
Q & A

  • What is the criterion for two functions to be inverses of each other?

    -Two functions, f(x) and g(x), are inverses of each other if the composition of the functions in both orders, f(g(x)) and g(f(x)), both equal x.

  • What is the first pair of functions discussed in the video script?

    -The first pair of functions discussed are f(x) = x^2 + 5 and g(x) = √(x - 5).

  • How does the video demonstrate that f(x) and g(x) are inverses for the first example?

    -The video shows that f(g(x)) simplifies to x and g(f(x)) also simplifies to x, confirming that they are inverses.

  • What is another method to verify if two functions are inverses?

    -Another method is to find the inverse function of f(x), denoted as f^(-1)(x), and check if it is the same as g(x).

  • What is the second pair of functions discussed in the video script?

    -The second pair of functions discussed are f(x) = 3x + 8 and g(x) = (8x^2 - 3)^(1/2).

  • Why are the functions f(x) = 3x + 8 and g(x) = (8x^2 - 3)^(1/2) not inverses of each other?

    -The functions are not inverses because f(g(x)) does not simplify to x, indicating that g(x) is not the inverse of f(x).

  • What is the third pair of functions discussed in the video script?

    -The third pair of functions discussed are f(x) = 7x + 5 and g(x) = (x - 5)/7.

  • How does the video show that f(x) = 7x + 5 and g(x) = (x - 5)/7 are inverses?

    -The video demonstrates that both f(g(x)) and g(f(x)) simplify to x, confirming that they are inverses.

  • What is the process of finding the inverse function of f(x)?

    -To find the inverse function, you replace x with y in the equation y = f(x), then solve for y.

  • What is the fourth pair of functions discussed in the video script?

    -The fourth pair of functions discussed are f(x) = 5/x + 3 and g(x) = 5/(x - 3).

  • How does the video demonstrate that f(x) = 5/x + 3 and g(x) = 5/(x - 3) are inverses?

    -The video shows that both f(g(x)) and g(f(x)) simplify to x, confirming that they are inverses.

  • What is the fifth pair of functions discussed in the video script?

    -The fifth pair of functions discussed are f(x) = 8x + 7 and g(x) = 7/x - 8.

  • Why are the functions f(x) = 8x + 7 and g(x) = 7/x - 8 not inverses of each other?

    -The functions are not inverses because f(g(x)) does not simplify to x, indicating that they do not cancel out to the identity function.

  • What is the sixth and final pair of functions discussed in the video script?

    -The sixth pair of functions discussed are f(x) = x^3 + 5 and g(x) = ∛(x - 5).

  • How does the video show that f(x) = x^3 + 5 and g(x) = ∛(x - 5) are inverses?

    -The video demonstrates that both f(g(x)) and g(f(x)) simplify to x, confirming that they are inverses.

Outlines
00:00
🔍 Verifying Inverse Functions Through Composition

This paragraph explains how to determine if two functions, f(x) and g(x), are inverses of each other. The process involves checking if the composition of the two functions, f(g(x)) and g(f(x)), both result in the original variable x. The example functions given are f(x) = x^2 + 5 and g(x) = √(x - 5). After substituting g(x) into f(x) and simplifying, it's shown that f(g(x)) = x. Similarly, substituting f(x) into g(x) confirms that g(f(x)) = x, proving that the functions are indeed inverses. An alternative method is also discussed, which involves finding the inverse function of f(x) by replacing x with y and solving for y, resulting in the same function g(x). This confirms that f(x) and g(x) are inverses.

05:00
📚 Analyzing Function Inverses with Algebraic Manipulation

The second paragraph explores whether different pairs of functions are inverses by algebraically manipulating their compositions. The first pair, f(x) = 3x + 8 and g(x) = 8x^2 - 3, is shown not to be inverses because f(g(x)) does not simplify to x. The second pair, f(x) = 7x + 5 and g(x) = (x - 5)/7, is confirmed to be inverses as both compositions f(g(x)) and g(f(x)) simplify to x. The process involves substituting one function into the other and simplifying to check for the identity. The inverse of f(x) is also calculated to verify that it matches g(x), reinforcing the conclusion that the functions are inverses.

10:02
📉 Examining Additional Examples of Function Inverses

This paragraph continues the theme of determining if given functions are inverses by examining additional examples. The first example involves functions f(x) = 5/x + 3 and g(x) = 5/(x - 3), where both compositions f(g(x)) and g(f(x)) simplify to x, confirming they are inverses. The second example features f(x) = 8x + 7 and g(x) = 7/x - 8, which are not inverses as f(g(x)) does not equal x. The final example presents f(x) = x^3 + 5 and g(x) = ∛(x - 5), which are shown to be inverses through the same composition process. Each example illustrates the method of substituting and simplifying to determine the relationship between the functions.

Mindmap
Keywords
💡Inverse Functions
Inverse functions are a mathematical concept where one function reverses the effect of another. Specifically, if you apply function 'f' to a number and then apply its inverse function 'g', you should get back the original number. In the video, the main theme revolves around determining whether two given functions are inverses by showing that f(g(x)) = x and g(f(x)) = x.
💡Function Composition
Function composition refers to applying one function to the result of another. It is denoted as f(g(x)) and is a key method to verify if two functions are inverses. In the script, function composition is used to test the inverse relationship by substituting one function into another.
💡Square Root
The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. In the context of the video, the square root is used in the definition of one of the functions, g(x) = sqrt(x - 5), to demonstrate an example of an inverse function.
💡Square
The square of a number is the result of multiplying the number by itself. In the script, squaring is used in the function f(x) = x^2 + 5, and it is shown that when composed with its inverse, the square operation is canceled out, resulting in the identity function.
💡Cube Root
The cube root of a number is a value that, when cubed, gives the original number. The video uses the cube root in the function g(x) = cuberoot(x - 5) to illustrate another example of an inverse function where the cube operation is reversed.
💡Cube
Cubing a number involves raising it to the power of three. In the script, the function f(x) = x^3 + 5 involves cubing, and it is demonstrated that its inverse function involves taking the cube root to reverse the cubing operation.
💡Distributive Property
The distributive property is a fundamental arithmetic principle that allows you to multiply a term by a sum by multiplying the term by each addend individually and then summing the products. In the script, the distributive property is implicitly used when expanding expressions like 3 * (8x^2 - 3).
💡Simplifying Expressions
Simplifying expressions involves reducing mathematical expressions to their simplest form. In the video, simplifying is crucial for determining if the compositions f(g(x)) and g(f(x)) result in the identity function, x.
💡Identity Function
The identity function is a function that always returns the same value that was used as its input. It is the simplest kind of function where f(x) = x. The video's theme of inverse functions hinges on the identity function, as the goal is to show that f(g(x)) = x and g(f(x)) = x.
💡Inverse Function Calculation
Calculating the inverse of a function involves algebraic manipulation to isolate the variable in terms of the output. In the script, the process of finding the inverse function is demonstrated by solving for y in terms of x after setting y = f(x), and then interchanging x and y.
💡Complex Fractions
Complex fractions are fractions that have other fractions in their numerator, denominator, or both. In the script, when dealing with functions like f(x) = 5/(x + 3), complex fractions arise during the composition process, and they are simplified to determine if the functions are inverses.
Highlights

Introduction to the concept of showing two functions are inverses of each other.

The method involves verifying if the composition of two functions equals the original variable x.

First example: f(x) = x^2 + 5 and g(x) = sqrt(x - 5) are tested for being inverses.

f(g(x)) simplifies to x, indicating a potential inverse relationship.

g(f(x)) also simplifies to x, confirming f and g are inverses.

An alternative method is to find the inverse function of f and compare it with g.

Second example: f(x) = 3x + 8 and g(x) = 8x^2 - 3 are not inverses as f(g(x)) does not equal x.

Third example: f(x) = 7x + 5 and g(x) = (x - 5)/7 are shown to be inverses.

Verification of inverse relationship by showing f(g(x)) and g(f(x)) both equal x.

Finding the inverse of f by algebraic manipulation confirms it matches g(x).

Practice problem presented: Determine if f(x) = 5/x + 3 and g(x) = 5/(x - 3) are inverses.

Multiplication strategy used to simplify f(g(x)) to x, indicating potential inverses.

g(f(x)) simplifies to x, confirming f and g are inverses for the practice problem.

Example with f(x) = 8x + 7 and g(x) = 7/x - 8 shows they are not inverses as f(g(x)) does not equal x.

Final example: f(x) = x^3 + 5 and g(x) = cube root(x - 5) are tested for inverse relationship.

f(g(x)) simplifies to x, and g(f(x)) also simplifies to x, confirming they are inverses.

Transcripts

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