Inverse functions 1.6
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The document discusses functions and their inverse functions. It provides an analogy that functions are like dye that colors eggs, and the inverse "undoes" the dye by bleaching the egg. The inverse of a function undoes what the original function did. For the square function f(x)=x^2, its inverse is the square root function. Graphically, the inverse function switches the x and y values of a point. The graphs of a function and its inverse are mirror images across the line y=x. Examples are provided to demonstrate finding the inverse of functions by switching x and y and solving for y.
Inverse functions 1.6
- 2. FunctionsFunctions Imagine functions are like the dye you useImagine functions are like the dye you use to color eggs. The white egg (x) is put into color eggs. The white egg (x) is put in the function blue dye B(x) and the result isthe function blue dye B(x) and the result is a blue egg (y).a blue egg (y).
- 3. The Inverse Function “undoes” what the functionThe Inverse Function “undoes” what the function does.does. The Inverse Function of the BLUE dye is bleach.The Inverse Function of the BLUE dye is bleach. The Bleach will “undye” the blue egg and make itThe Bleach will “undye” the blue egg and make it white.white.
- 4. In the same way, the inverse of a givenIn the same way, the inverse of a given function will “undo” what the originalfunction will “undo” what the original function did.function did. For example, let’s take a look at the squareFor example, let’s take a look at the square function: f(x) = xfunction: f(x) = x22 33 xx f(x)f(x) 3333333333 99999999999999 yy ff--1--1 (x)(x) 99999999999999 33333333333333 x2 x
- 5. 555555555555 25252525 2525 2525 25252525 2525 25252525252555 5555555555555555 In the same way, the inverse of a givenIn the same way, the inverse of a given function will “undo” what the originalfunction will “undo” what the original function did.function did. For example, let’s take a look at the squareFor example, let’s take a look at the square function: f(x) = xfunction: f(x) = x22 xx f(x)f(x) yy ff--1--1 (x)(x) x2 x
- 6. 111111111111111111111111 121121121121121121121121121121121121121121121121121121121121121121121121121121121121 11111111111111111111111111111111 In the same way, the inverse of a givenIn the same way, the inverse of a given function will “undo” what the originalfunction will “undo” what the original function did.function did. For example, let’s take a look at the squareFor example, let’s take a look at the square function: f(x) = xfunction: f(x) = x22 xx f(x)f(x) yy ff--1--1 (x)(x) x2 x
- 7. Graphically, the x and y values of aGraphically, the x and y values of a point are switched.point are switched. The point (4, 7)The point (4, 7) has an inversehas an inverse point of (7, 4)point of (7, 4) ANDAND The point (-5, 3)The point (-5, 3) has an inversehas an inverse point of (3, -5)point of (3, -5)
- 8. -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 Graphically, the x and y values of a point are switched.Graphically, the x and y values of a point are switched. If the function y = g(x)If the function y = g(x) contains the pointscontains the points then its inverse, y = gthen its inverse, y = g-1-1 (x),(x), contains the pointscontains the points xx 00 11 22 33 44 yy 11 22 44 88 1616 xx 11 22 44 88 1616 yy 00 11 22 33 44 Where is there aWhere is there a line of reflection?line of reflection?
- 9. The graph of aThe graph of a function andfunction and its inverse areits inverse are mirror imagesmirror images about the lineabout the line y = xy = xy = f(x)y = f(x) y = fy = f-1-1 (x)(x) y = xy = x
- 10. Find the inverse of a function :Find the inverse of a function : Example 1:Example 1: y = 6x - 12y = 6x - 12 Step 1: Switch x and y:Step 1: Switch x and y: x = 6y - 12x = 6y - 12 Step 2: Solve for y:Step 2: Solve for y: x = 6y −12 x +12 = 6y x +12 6 = y 1 6 x + 2 = y
- 11. Example 2:Example 2: Given the function :Given the function : y = 3xy = 3x22 + 2+ 2 find the inverse:find the inverse: Step 1: Switch x and y:Step 1: Switch x and y: x = 3yx = 3y22 + 2+ 2 Step 2: Solve for y:Step 2: Solve for y: x = 3y2 + 2 x − 2 = 3y2 x − 2 3 = y2 x − 2 3 = y