Understanding inverse functions

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Discussion Overview

The discussion revolves around understanding inverse functions, particularly how to calculate them and apply them to specific examples. Participants are exploring the definitions and properties of inverse functions, as well as methods for solving related problems, including finding specific values of inverse functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant expresses confusion about calculating the inverse of functions, specifically asking for help with two examples from their textbook.
  • Another participant suggests visualizing the function and its inverse by drawing the function and its reflection across the line y=x, and discusses how shifting the function affects its inverse.
  • A third participant clarifies the meaning of g-1(y) and explains that it involves finding an x such that g(x) equals a given value, leading to a specific equation to solve.
  • In the first example, a participant outlines the steps to derive g-1(x) from g(x) by manipulating the equation and applying the definition of inverse functions.
  • For the second example, a participant provides a method to find g-1(1) by solving the equation g(x)=1 and suggests using the rational root theorem to identify potential solutions.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and methods for calculating inverse functions, but there is no consensus on the best approach to explain these concepts clearly to someone struggling with the material.

Contextual Notes

Some participants note the importance of understanding the technical terms involved in discussing inverse functions, which may be challenging for non-native speakers. There are also references to specific mathematical techniques, such as the rational root theorem, which may require further clarification for some participants.

Who May Find This Useful

This discussion may be useful for students learning about inverse functions, particularly those seeking clarification on how to apply the concepts in problem-solving contexts. It may also benefit individuals looking for different perspectives on teaching or understanding mathematical concepts.

admiralgman
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I need some help understanding inverse functions, we've had a 4-page chapter covering the basics of inverse functions and I understand that.

But now we have suddenly gotten these task that I don't understand how to solve, I've read the part on inverses several times, but I still don't understand.

To explain my trouble, here are two examples from my textbook that covers what I am struggeling with:

1:Calculate the inverse of the given functions in terms of f-1

g(x)=f(x)-2

(answer to this should be g-1(x)=f-1(x+2))

2: Find g-1(1) if g(x)=x3+x-9

(answer to this should be 2)


As you see I know the answers, this because they are in the back of my textbook. But I want some help understanding how you get from that point A to B
Also, it would be nice if you could show it in a step-by-step way as I am (already) having some trouble understanding the technical terms in english, considering I am from Norway

As a last bit of information; I will be gone for two hours or so now(Im a busy man), but it would be nice to come back here and see something that would help me understand "advanced" inverses.
 
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admiralgman said:
1:Calculate the inverse of the given functions in terms of f-1

g(x)=f(x)-2

(answer to this should be g-1(x)=f-1(x+2))

For this one, it would probably help if you drew some random function f(x) and to keep things simple, have it be entirely below the line y=x. Draw a dotted line y=x for reference, and then draw the inverse function g(x) = f-1(x) which is a reflection of f(x) in the line y=x.

Now, if you draw f(x)-2, that's simply a shift of f(x) down 2 units, and so if you reflect f(x)-2 across the y=x line (which is also the same as considering what the inverse function is) then how is this new function shifted in relation to g(x)? Is it shifted down, up, left, right?

admiralgman said:
2: Find g-1(1) if g(x)=x3+x-9

(answer to this should be 2)

If you were supposed to find g(2), then what do you get?
 
admiralgman said:
2: Find g-1(1) if g(x)=x3+x-9

(answer to this should be 2)
I'm only going to help with this second example as Mentallic already gave a nice way to look at the first example.

Rhetorical question: What does g-1(y) mean? The y is some value in the range (image) of g(x), and g-1(y) maps from this range back to the domain of g(x) in such a way that g-1(g(x))=x and g(g-1(y))=y.

What you are looking for here is some x such that g(x)=1. In other words, you need to solve x3+x-9=1, or x3+x-10=0. Solving this isn't that hard. Just use the rational root theorem. The only possible rational roots are ±10, ±5, ±2, and ±1. You should be able to eliminate all but ±2 by inspection. x-2 divides x3+x-10, so the solution is x=2.
 
admiralgman said:
I need some help understanding inverse functions, we've had a 4-page chapter covering the basics of inverse functions and I understand that.

But now we have suddenly gotten these task that I don't understand how to solve, I've read the part on inverses several times, but I still don't understand.

To explain my trouble, here are two examples from my textbook that covers what I am struggeling with:

1:Calculate the inverse of the given functions in terms of f-1

g(x)=f(x)-2
By the definition of "inverse function" if y= f(x) then x= f-1(y). Similarly, if y= g(x) then x= g-1(x).

Let y= g(x)= f(x)- 2. Then f(x)= y+ 2 so that x= f-1(y+ 2). Writing that in terms of the variable x, g-1(x)= f-1(x+ 2)

(answer to this should be g-1(x)=f-1(x+2))

2: Find g-1(1) if g(x)=x3+x-9
If g(x)= x3 + x- 9, then g(2)= 23+ 2- 9= 8+ 2- 9= 1
so that g-1(1)= 2.

(answer to this should be 2)


As you see I know the answers, this because they are in the back of my textbook. But I want some help understanding how you get from that point A to B
Also, it would be nice if you could show it in a step-by-step way as I am (already) having some trouble understanding the technical terms in english, considering I am from Norway

As a last bit of information; I will be gone for two hours or so now(Im a busy man), but it would be nice to come back here and see something that would help me understand "advanced" inverses.
 

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