The Way Of The Inverse!


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The Way Of The Inverse!

  1. 1. The Way of the Inverse! By Sean Prins For: Education 205 End Show
  2. 2. Inverse’s <ul><li>What is an inverse? </li></ul><ul><ul><li>Examples of how to get an inverse. </li></ul></ul><ul><ul><li>Graphing inverse! </li></ul></ul><ul><ul><li>Video to help you graph inverse! </li></ul></ul><ul><li>Definition for two inverses being inverses of each other. </li></ul><ul><ul><li>Examples of two inverses being inverses of each other . </li></ul></ul><ul><li>Authors page </li></ul><ul><li>Resource page </li></ul><ul><li>Concept map </li></ul>End Show
  3. 3. What is an inverse <ul><li>When you are trying to find an inverse for an equation, you have to switch the x’s with the y’s and the y’s with the x’s. </li></ul><ul><li>- Example one </li></ul><ul><li>- Example two </li></ul>End Show
  4. 4. Example 1 <ul><li>So you have the equation. </li></ul><ul><ul><li>y=2x-5 </li></ul></ul><ul><li>First you switch the x and the y. </li></ul><ul><ul><li>x=2y-5 </li></ul></ul><ul><li>Then you solve for y. </li></ul><ul><ul><li>x=2y-5 First add 5 on both sides. </li></ul></ul><ul><ul><li>x+5=2y-5+5 The 5’s on the right side cancel out. </li></ul></ul><ul><ul><li>x+5=2y Multiply by ½ on both sides. </li></ul></ul><ul><ul><li>½(x+5)=(2y)1/2 The 2 and the ½ cancel out on the right side </li></ul></ul><ul><ul><li>(x+5)/2=y And that is the inverse! </li></ul></ul><ul><ul><li>Click here for example 2 </li></ul></ul>End Show
  5. 5. Example 2 <ul><li>So you have the equation. </li></ul><ul><ul><li>y=x ² +5 </li></ul></ul><ul><li>First you switch the x and the y. </li></ul><ul><ul><li>x=y ² +5 </li></ul></ul><ul><li>Then you solve for y. </li></ul><ul><ul><li>x=y ² +5 Subtract 5 from both sides </li></ul></ul><ul><ul><li>x-5=y ² +5-5 The 5 by the y cancel out </li></ul></ul><ul><ul><li>x-5=y ² Then square root both sides </li></ul></ul><ul><ul><li>√ x-5 = √y² The squared and the square root cancel out. </li></ul></ul><ul><ul><li>√ x-5=y There you have it, that’s the answer! </li></ul></ul>End Show
  6. 6. Graph of the inverse. <ul><li>I am going to use the example that I used in example one. </li></ul><ul><li>The original equation is in red and the inverse is in black. </li></ul><ul><li>Do you see a connection between the two functions? </li></ul><ul><li>Click here to find out the </li></ul><ul><li> answer! </li></ul>End Show
  7. 7. Answer <ul><li>Yes these is a connection. The inverse is flipped over the y=x axis. </li></ul><ul><li>But just because the inverse can be graphed, it does not mean it is a function. You still have to do the Horizontal Line Test. So the graph on the left would fall the HLT. </li></ul>End Show
  8. 8. Video of Graphing! <ul><li>Click here for the video of solving and graphing an inverse function. This should be really good! </li></ul>End Show
  9. 9. How do you check to see if two graphs are inverses of each other? <ul><li>You will be given to equations. Lets say you are given f(x) and g(x). The only way you can tell they are inverses is if you take one and substitute it for the other. For instance, you need to take g(x) and put it into f(x). This is what it looks like. f(g(x)). Then you have to do the opposite. g(f(x)). The only way they are inverses is if f(g(x)) and g(f(x)) equals x. </li></ul><ul><li>Example 1 </li></ul><ul><li>Example 2 </li></ul>End Show
  10. 10. Example 1 <ul><li>Lets use equations that we already know are inverses. </li></ul><ul><li>f(x)=2x-5 and g(x)=(x+5)*1/2. </li></ul><ul><li>f(g(x))=2((x+5)*1/2)-5 The 2 and ½ cancel out. </li></ul><ul><li>f(g(x))=(x+5)-5 The 5 and -5 cancel out. </li></ul><ul><li>f(g(x))=x </li></ul><ul><li>Now for g(f(x)) </li></ul><ul><li>g(f(x))=((2x-5)+5)* ½ The 5 cancel out leaving </li></ul><ul><li>g(f(x))=(2x)*1/2 The 2 and ½ cancel out </li></ul><ul><li>g(f(x))= x </li></ul><ul><li>g(f(x))=f(g(x)) They are inverses. </li></ul><ul><li>Click here for another example! </li></ul>End Show
  11. 11. Example 2 <ul><li>Now for my second example, I am going to give you the problems and you are going to have to try to solve it. The answer will be in the link below, only if you have any trouble with it. </li></ul><ul><li>f(x)=x³+8 </li></ul><ul><li>g(x)=³√(x)-2 </li></ul><ul><li>Answer here! </li></ul>End Show
  12. 12. Answer <ul><li>Yes they are inverses of each other. </li></ul><ul><li>f(x)=x³+8 and g(x)=³√(x)+2 </li></ul><ul><li>f(g(x))= (³√(x)-2)³+8 You have to distibute the third power to the ³√(x) and the 2. </li></ul><ul><li>f(g(x))= ³√(x) ³-(2) ³+8 The third power and the ³√ cancel out. </li></ul><ul><li>f(g(x))=x-8+8 </li></ul><ul><li>f(g(x))=x </li></ul><ul><li>Click here for g(f(x ))! </li></ul>End Show
  13. 13. Answer 2 for example 2! <ul><li>f(x)= x³+8 and g(x)=³√(x)-2. This is what you need to do for g(f(x)). </li></ul><ul><li>g(f(x))=³√(x³+8)-2 This is what you need to start out with. </li></ul><ul><li>g(f(x))=³√(x³)+ ³√(8)-2 You have to distibute the cube root to x³ and the 8. </li></ul><ul><li>g(f(x))=x+2-2 Cancel out the cube rood and the third power. </li></ul><ul><li>g(f(x))=x g(f(x)) equals the same as f(g(x)). They are inverses! </li></ul>End Show
  14. 14. Authors Page <ul><li>Sean Prins is a student at Grand Valley </li></ul><ul><li>State University. He is currently trying </li></ul><ul><li>to get his major in Mathematics and </li></ul><ul><li>has not chosen a minor yet. He tries </li></ul><ul><li>to be very active. He plays hockey </li></ul><ul><li>almost every week and enjoys fishing </li></ul><ul><li>on the weekends. If he is not in school, like during the summer and winter break, he is working many hours a week at his job. </li></ul>End Show
  15. 15. Resource slide <ul><li> </li></ul><ul><li> </li></ul><ul><li> </li></ul><ul><li> </li></ul>End Show
  16. 16. Concept Map End Show