1. Parametric and Inverse Functions Nate Hutnik Aidan Lindvig Craig Freeh

2. Intro Video

3. Defining a Function Parametrically Parameter: Another way to define functions by defining the X and Y in terms of variable T. (x = t + 1) (y = t 2 + 2t) Now say that t=2, then just plug the number 2 into each equation to get the x and y coordinates. They are (3,8) The relationship can also be solved with substitution. Solve for t in terms of x. t = x - 1. Then plug that into the y equation. Then you end up with y = x 2 - 1, which is the same as just solving for each point with t. Tips: a graphing calculator can be used to graph it parametrically, it gives an x= and a y= to enter in the relationship of t.

4. Inverse relations and functions Inverse relation: The ordered pair (a,b) is in a relation if and only if the ordered pair (b,a) is in it. Horizontal line test: We've heard of the vertical line test, so the horizontal line test simply test the first equation, so see if the inverse will be a function or not. The intial graph failed the horizontal line test, so the inverse that opens to the right failed the vertical test. Inverse is f -1

5. Finding Inverse Algebraically Simply switch the x's and y's, and then solve it again for y. Note the domain and range. x = y/y+1 Example x(y+1) = y multiply by y+1 xy + x = y distribute xy - y = -x isolate y y(x-1) = -x factor out y y = -x/x-1 divide by x-1 y = x/1-x multiply top and bottom by -1

6. Inverse with reflections The points (a,b) and (b,a) are reflections of across the line y=x. they both have inverses because they passed the horizonal line test.

7. Verifying Inverse Functions Inverse rule: A function is an inverse if f(g(x) = x and g(f(x) = x. Example- f(x)=3x-2 g(x)=x+2/3 f(g(x) = 3(x+2/3)-2 = x+2-2 = x g(f(x) = 3x-2+2/3 = 3x/3 = x

8. Examples Find the value of the parameter: x = 4t and y = t 2 + 5 for t=2 x = 4(2) x = 8 y = (2) 2 + 5 y = 4 + 5 y = 8 (8,9) Points can each be found like this..... or Take previous two parameters and make an equation of y in terms of x. t=x/4 y=(x/4) 2 + 5 y = (x 2 /16) +5...either way works, and the same relationship is there.

9. Examples (a) is it a function (b) is its inverse a function (a) yes, because it passes the vertical line test. (b) no, because it failes the horizontal line test.