Verifying Inverses Verifying –checking that something is true or correct Inverses – two equations where the x’s and y’s of one equation are switched in.

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Presentation transcript:

Verifying Inverses Verifying –checking that something is true or correct Inverses – two equations where the x’s and y’s of one equation are switched in the second equation

How do we verify inverses?  There are two ways to determine if two functions are inverses of each other: 1. Graphically – Are all the points of one graph reflections over the line y=x from the other graph? (Corresponding points have to switch the x and y coordinates.) 2. Algebraically – Do both compositions simplify to “x”?

Verifying Inverses GRAPHICALLY Is EVERY point on one graph a reflection over the line y = x of a point on the other graph? Do they cross the line y = x at the same place? Yes, they are INVERSES since all points are reflections over y = x !!! And they intersect on the “mirror” line.

Verifying Inverses GRAPHICALLY Are the lines reflections over y = x? Do they intersect on the line y = x? No. They do not intersect on the “mirror” line. Nor do the points reflect over y = x! Ex: (4,10) does not go to (10,4)

Verifying Inverses ALGEBRAICALLY  If functions f and g are inverses of each other, then f(g(x)) = x and g(f(x)) = x.  What this means is: if we substitute one function’s equation in for the x of the other equation and then simplify the expression, we’ll be left with just ‘x’. *** f(g(x)) and g(f(x)) are called compositions of functions.

Verifying Inverses ALGEBRAICALLY Find both compositions… Plug each function into the other and simplify. f(g(x)) = 2(1/2x – 2) + 4 = x – … distribute = x … this one is good. g(f(x)) = ½ (2x + 4) – 2 = x + 2 – 2… distribute = x … both are good Since both compositions simplify to “x” they are inverses of each other.

Extra Example Find both compositions… Plug each function into the other and simplify. f(g(x)) = 4(1/4x + 4) – 16 = x + 16 – 16 … distribute = x … this one is good. g(f(x)) = ¼ (4x – 16) + 4 = x – 4 + 4… distribute = x … both are good Since both compositions simplify to “x” they are inverses of each other.

Verifying Inverses Algebraically and Graphically f(g(x)) = 4(1/4x – 3) + 3 = x – … distribute = x – 9 … this one is NOT good so they cannot be inverses g(f(x)) = ¼ (4x + 3) – 3 = x + 3/4 – 3 = x – 2.25 … neither one is good If either one of the compositions does NOT simplify to “x” then they are NOT inverses of each other. GRAPHICALLY… They do NOT intersect on the line y = x so they are NOT inverses. (Points are NOT reflected across y = x either.)

Verifying Inverses Graphically and Algebraically Try #5 on your own, and then you can check your work by moving on to the next two slides.  The first slide will show the problem done algebraically.  The second slide will show the problem done graphically.

#5 –Algebraically

#5 - Graphically