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### Course: Get ready for Precalculus>Unit 3

Lesson 6: Intro to inverse functions

# Graphing the inverse of a linear function

Sal is given a line segment on the coordinate plane, and he graphs the inverse of the function represented by that segment.

## Want to join the conversation?

• Are all linear functions invertible around y=x?
• All functions are at least in part invertible (for some functions you have to only take a small part of the function in order for it to be invertible), but once you invert them, graphically they all look like they were mirrored about the line `y = x`.
• Sal said in his first video in inverse function that he will explain why he constrained the domain of function. Where is the video that explain the reason ?
• In Algebra 1, the first module is functions. Check it up, it's there
(1 vote)
• Will the line Y=X stay the same or would something strange happen?
• Can someone please explain to me the concept of the horizontal line test? I'm doing the domain and range with inverse functions exercise.
• The horizontal line test is used for figuring out whether or not the function is an inverse function. Picture a upwards parabola that has its vertex at (3,0). Then picture a horizontal line at (0,2). The line will touch the parabola at two points. This is how you it's not an inverse function.
• how to do the inverse of trigonometric functions?
(1 vote)
• how do you solve the inverse function of y=x^2 +2
• The inverse can be found by switching the x and y, then solving for y again.

So the inverse is x=y^2 +2
x-2=y^2
y=√(x-2)
• What is that -1 exponent up there on the function? I know it means ‘inverse’ but is there some other explanation as to why it looks like that?
• Well, imagine if we could just slap a '-2' on any function and magically reverse it twice! We'd be undoing the undo, then redoing the redo, and things would get pretty wild. It's like trying to rewind a movie you're already rewinding! So, instead of diving into that infinite loop, mathematicians kept it simple with just a '-1' for the inverse, less headache. The "-1" notation is a convention used specifically for denoting inverse functions.
• Implicitly defined curves do have "inverses" but I don't understand why they may not be the graph of a function. For example, if I plot the inverse relation of f(x) = x^3-4x, how can you say it is not the graph of a function? It would really help me out if you could answer this question. Thank you : )
• So the function you present has an inverse that is not a function. If you factor this out to x(x^2-4) or x(x-2)(x+2) you have three x intercepts at (0,0) (2,0) and (-2,0). If you switched x and y, your inverse function would have 3 y intercejpts at (0,0) (0,2) and (0,-2) which does not pass the vertical line test and has one x value go to 3 different y values, and is thus not a function. Some cubic functions such as the parent function (y=x^3) or with just a cube and constant (y=x^3 - 27) would have inverses that are functions.
By starting out with a function, you know it passes the vertical line test by definition. However, for its inverse to be a function, it also has to pass the horizontal line test to insure the one-to-one correspondence of x and y values.