sat problem of the day. 2.5 inverses of functions 2.5 inverses of functions objectives: find the...
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SATProblem of the Day
2.5 Inverses of Functions2.5 Inverses of Functions2.5 Inverses of Functions2.5 Inverses of FunctionsObjectives: •Find the inverse of a relation or function•Determine whether the inverse of a function is a function
Example 1Solve the equation v = 50 + 3t for t.
1 50
t v3 3
v = 50 + 3t-50 -50
v - 50 = 3t 3 3
Inverse of a Relation
The domain of the inverse is the range of the original relation.
The range of the inverse is the domain of the original relation.
The inverse of a relation consisting of the ordered pairs (x, y) is the set of all ordered pairs (y, x).
Example 2Find the inverse of each relation. State whether the relation is a function. State whether the inverse is a function.
a) {(1,2), (4,-2), (3,2)}inverse: {(2,1), (-2,4), (2,3)}
functionnot a function
b) {(-2,4), (3,-4), (-8,-5)}inverse: {(4,-2), (-4,3), (-5,-
8)}
functionfunction
Example 3Find an equation for the inverse of .
2x 3y
5
2y 3x
5
2y 3
x5
5 5
interchange x and ysolve for y
5x 2y 3 solve for y-3 -35x - 3 = 2y
2 2
5 3
y x2 2
PracticeFind an equation for the inverse of
f(x) = 4x – 5.
Activity1) Graph y = 2x – 1.2) Graph y = x.
3) Graph the inverse of y = 2x – 1.
4) Graph y = -2x + 55) Graph y = x.
6) Graph the inverse of y = -2x + 5.
Graphs of Inverse Functions
The graph of the inverse of a function is the reflection of the graph of the function across the line y = x.
Horizontal-Line TestThe inverse of a function is a function iff every horizontal line intersects the graph of the given function at no more than one point.
Horizontal-Line Test
not a function function
Composition and Inverses
If f and g are functions and (f ○ g)(x) = (g ○ f)(x) = x then f and g are inverses of one another.
Example 4Show that f(x) = 4x – 3 and are inverses of each other.
1 3
g(x) x4 4
(f g)(x) f(g(x))
1 3x
4 4f
1 34 x 3
4 4
x 3 3
(g f )(x) g(f(x))
g(4x 3)
1 3
(4x 3)4 4
3 3
x4 4
x x
Since , the two functions are inverses of each other.
(f g)(x) (g f )(x) x
PracticeShow that f(x) = -5x + 7 and are inverses of each other.
1 7
g(x) x5 5
Homework
More Problems on Inverses
Collins WritingType 1
Compare and contrast the vertical- and horizontal-line tests.