ch. 5.5 building inverses of functions - brenegan's...

Ch. 5.5 Building Inverses of FunctionsLearning Intentions:

Define the inverse relation of a function.

Given several points in a function, find inverse points.

See relation-inverse symmetry across the line y = x.

Find the composition of functions with their inverses.

Apply inverses in real-world situations.

± +

5 3

-3

5.5 Building Inverses of FunctionsINVESTIGATION:Inverse Function: where the independent variable is exchanged with the dependent variable.f(x) = 3x + 6 g(x) = _________

Write an equation for g(x).How are these tables related?

How is f(x) related to g(x)?

x f(x)

-2 0

-1 3

0 6

1 9

2 12

x g(x)

0 -2

3 -1

6 0

9 1

12 2

SOLUTIONS: 5.5 Building Inverses of Functions INVESTIGATION

Inverse Function: where the independent variable is exchanged with the dependent variable.

f(x) = 3x + 6 g(x) = 𝟏

𝟑x – 2

Write an equation for g(x).x f(x)

-2 0

-1 3

0 6

1 9

2 12

x g(x)

0 -2

3 -1

6 0

9 1

12 2

For y = mx + bSince (0, b) = (0, -2)b = -2

m = 𝑦2 − 𝑦1

𝑥2 − 𝑥1Let (𝑥1, 𝑦1) = (3, -1)& (𝑥2, 𝑦2) = (6, 0)

m = 0 −(−1)

6 −3= 𝟏

𝟑Thus,

g(x) = 𝟏

𝟑x – 2

How are these tables related?The x & y-values are switched.(the independent variable becomes the dependent variable in the other table).

How is f(x) related to g(x)?g(x) is the inverse equation of f(x) andf(x) is the inverse equation of g(x).

A function & itsinverse are REFLECTIONSacross the line y = x.

One-to-One Function: when an equation and its inverse are both functions.How can you tell?Your original function has no more than one x-value for every y-value. A function and its inverse are reflections across the line y = x. If the reflection of your function passes the vertical line test, you have a one-to-one function.OR – your original function passes both a vertical AND horizontal line

test, the function AND its inverse are 1-to-1.

Finding the Composition of a function f(x) with its inverse 𝑓−1(x):Finding f(𝑓−1(x)) = ________ OR 𝑓−1(f(x)) = ________ STEPS:1.) Find the inverse of the given function (exchange x & y, solve for y).

𝑓−1(x) = ______2.) Form the composition of the two functions.

- substitute 𝑓−1(x) for x in f(x) - substitute f(x) for x in 𝑓−1(x) f(𝑓−1(x)) = ________ OR 𝑓−1(f(x)) = ________

3.) Both f(𝑓−1(x)) AND 𝑓−1(f(x)) should equal x.

Ex.) Find the composition of f(x) & its inverse given f is 𝒙𝟐 + 2y = 3.If the inverse of f(x) is a function, write 𝑓−1(x). If not, write y.

Finding the Composition of a function f(x) with its inverse 𝑓−1(x):Finding f(𝑓−1(x)) = ________ OR 𝑓−1(f(x)) = ________ 1.) Find the inverse of the given function (exchange x & y, solve for y).2.) Form the composition of the two functions.

- substitute 𝑓−1(x) for x in f(x) - substitute f(x) for x in 𝑓−1(x) f(𝑓−1(x)) = ________ OR 𝑓−1(f(x)) = ________

3.) Both f(𝑓−1(x)) AND 𝑓−1(f(x)) should equal x.

Ex.) Find the composition of f(x) & its inverse given 𝒙𝟐 + 2y = 3.If the inverse of f(x) is a function, write 𝑓−1(x). If not, write y.

If 𝒙𝟐 + 2y = 3 then y = −𝟏

𝟐𝒙𝟐 + 1.5 which is a function: f(x) =

−𝟏

𝟐𝒙𝟐 + 1.5

The inverse of f(x) is: (replace x with y and y with x)

𝒚𝟐 + 2x = 3 where y = ± −𝟐𝒙 + 𝟑 which is NOT a function.Composition of f(x) & its inverse:

f(y) = f(± −𝟐𝒙 + 𝟑) …. simplify

f(y) = −𝟏

𝟐(± −𝟐𝒙 + 𝟑)𝟐 + 1.5

f(y) = −𝟏

𝟐(−𝟐𝒙 + 𝟑) + 1.5

f(y) = x – 1.5 + 1.5f(y) = x

Test your knowledge: p.284 #(3, 4)

SOLUTIONS: Test your knowledge: p.284 #(3, 4)

Graph c is the inverse because the x- and y-coordinates have been switched from the original graph so that the graphs are symmetric across the line y = x.