Verifying InversesVerifying Inverses

VerifyingVerifying –checking that something is –checking that something is true or correcttrue or correct

InversesInverses – two equations where the x’s – two equations where the x’s and y’s of one equation are switched and y’s of one equation are switched in the second equationin the second equation

How do we verify How do we verify inverses?inverses?

There are two ways to determine if two There are two ways to determine if two functions are inverses of each other:functions are inverses of each other:

1. Graphically – 1. Graphically – Are all the points of one graph Are all the points of one graph reflectionsreflections over the line over the line y=xy=x from the other graph? from the other graph? (Corresponding points have to switch the x and y coordinates.)(Corresponding points have to switch the x and y coordinates.)

2. Algebraically – 2. Algebraically – Do both Do both compositionscompositions simplify to “x”? simplify to “x”?

VerifyingVerifying Inverses Inverses GRAPHICALLYGRAPHICALLY

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 Verifying Inverses GRAPHICALLYGRAPHICALLY

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 Verifying Inverses ALGEBRAICALLYALGEBRAICALLY

If functions f and g are inverses of If functions f and g are inverses of each other, each other,

then then f(g(x)) = x f(g(x)) = x andand g(f(x)) = x g(f(x)) = x.. What this means is: What this means is:

if we if we substitute one function’s equationsubstitute one function’s equation in in for the xfor the x of the other equation and then simplify the of the other equation and then simplify the expression, we’ll be left with just ‘x’. expression, we’ll be left with just ‘x’.

*** f(g(x)) *** f(g(x)) and and g(f(x)) g(f(x)) are called are called compositionscompositions of of functions.functions.

Verifying Inverses Verifying Inverses ALGEBRAICALLYALGEBRAICALLY

Find both compositions… Plug each function into the other and simplify.

f(g(x)) = 2(1/2x – 2) + 4

= x – 4 + 4 … 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 ExampleExtra 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 Verifying Inverses Algebraically and Algebraically and GraphicallyGraphically

f(g(x)) = 4(1/4x – 3) + 3

= x – 12 + 3 … 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 Verifying Inverses Graphically and Graphically and AlgebraicallyAlgebraically

Try #5 on your ownTry #5 on your own, and then you can , and then you can check your work by moving on to the check your work by moving on to the next two slides. next two slides.

The first slide will show the problem The first slide will show the problem done algebraically.done algebraically.

The second slide will show the The second slide will show the problem done graphically.problem done graphically.

#5 –Algebraically#5 –Algebraically

#5 - Graphically#5 - Graphically