TLDR;
This video explains how to graph inverse functions by reflecting the original function across the line y = x. It covers determining the domain and range of inverse functions, and how they relate to the domain and range of the original function. The video includes examples with linear, rational, and cube root functions, and concludes with problem-solving examples and practice exercises.
- Graphing inverse functions involves reflecting the original function across the line y = x.
- The domain of the inverse function is the range of the original function, and vice versa.
- The video provides examples with linear, rational, and cube root functions.
Introduction to Inverse Functions [0:10]
The video introduces the concept of graphing inverse functions. Given a one-to-one function, its inverse can be graphed by reflecting the original graph across the line y = x. The presenter mentions using applications like Desmos to visualize this reflection.
Graphing the Inverse of a Linear Function [0:53]
The video demonstrates graphing the inverse of the linear function y = 2x + 1, restricted to a domain of -2 ≤ x ≤ 1.5. First, the original function is graphed within the specified domain. The range of this restricted function is then determined by observing the graph, which spans from -3 to 4. The reflection of this graph across the line y = x yields the inverse function. The domain of the inverse function is the range of the original function (-3 ≤ x ≤ 4), and the range of the inverse function is the domain of the original function (-2 ≤ y ≤ 1.5). The presenter emphasizes that the domain and range interchange between the original function and its inverse.
Domain and Range Relationship [5:33]
The video explains the relationship between the domain and range of a function and its inverse. The domain of the original function becomes the range of the inverse function, and the range of the original function becomes the domain of the inverse function. This interchange is a key property of inverse functions. The presenter then verifies this relationship using techniques for finding the inverse algebraically.
Inverse of f(x) = 1/x [7:48]
The video explores the inverse of the function f(x) = 1/x. The graph of 1/x is symmetric with respect to the line y = x, meaning its reflection across this line is itself. This implies that the inverse of f(x) = 1/x is also 1/x. Consequently, the domain and range of this function are the same. The presenter demonstrates algebraically how interchanging x and y in the equation results in the same function.
Inverse of a Cube Root Function [9:36]
The video demonstrates finding the inverse of the cube root function f(x) = ∛(x + 1). The original function is graphed, and its reflection across the line y = x is shown. The resulting inverse function is y = x³ - 1.
Inverse of a Rational Function [11:16]
The video explains how to find the domain, range, and asymptotes of a rational function and its inverse, using the example f(x) = (5x - 1) / (-x + 2). The domain of the original function is all real numbers except x = 2, as this value makes the denominator zero. The range is all real numbers except y = -5, found by dividing the leading coefficients of the numerator and denominator. For the inverse function, the domain and range are interchanged. The vertical asymptote of the original function (x = 2) becomes the horizontal asymptote of the inverse, and the horizontal asymptote of the original function (y = -5) becomes the vertical asymptote of the inverse.
Problem Solving Using Inverse Functions [16:59]
The video presents a word problem: "Pick a non-negative number, add 2, square the result, multiply by 3, and divide by 2. If the result is 54, what is the original number?" The problem is solved by constructing an inverse function. The original function is f(x) = (3(x + 2)²) / 2. By interchanging x and y and solving for y, the inverse function is found to be f⁻¹(x) = √(2x/3) - 2. Evaluating this inverse function at x = 54 yields the original number, which is 4.
Application of Inverse Functions [21:48]
The video presents an application problem: the maximum force T (in tons) that a particular bridge can withstand is related to the distance D (in meters) between its supports by the function T(D) = 12.5 / D³. The problem asks how far apart the supports should be if the bridge needs to support 6.5 tons. The inverse function is derived as D = ∛(12.5 / T). Evaluating this at T = 6.5 gives D ≈ 1.24 meters.
Practice Exercises and Solutions [23:22]
The video concludes with practice exercises.
- Exercise 1: Asks for the domain of the inverse of f(X) = √(4-X)
- Exercise 2: Asks to sketch the graph of its inverse
- Exercise 3: Asks to construct the universe of f(X) = √(4-X) using algebraic methods.
The domain of the inverse of f(x) is just the range of f(x) such that the range of f(x) is y or y is greater than 1 but less than or equal to 3 but therefore the domain in the inverse is x is greater than 1 but less than or equal to 3 in the solution for the practice exercises number 2 this the graph is the inverse of the original graph and for practice exercises number 3 so we're using this given function and we follow the steps in getting the inverse of this function so therefore the inverse is 4 minus x squared so ever the graph of that elicits a result a parabola so therefore the practice exercises number 3 is not a one-to-one it fails on the result align test ok that's all for the graphic of inverse function see guramukh gaga upon it come in another video for that but i must maintain the any path thank you so much and don't forget to Like subscribe and hit the bell boat on to our Walmart channel