7.4 Inverse Functions

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7.4 Inverse Functions

• p. 422

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Review from chapter 2

• Relation a mapping of input values (x-values)
  onto output values (y-values).
• Here are 3 ways to show the same relation.

x y -2 4 -1 1 0 0 1 1
y x2
Equation Table of values Graph
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• Inverse relation just think switch the x
  y-values.

• x y
• -2
• -1
• 0 0
• 1 1

y x2
the inverse of an equation 1) solve for x,
2) switch the x y.
the inverse of a table switch the x y.
the inverse of a graph the reflection of the
original graph in the line y x.
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• Function a mapping of input values (x-values,
  domain) onto output values (y-values, range) AND
  any one input is mapped to one and only one
  output.

• Function is a relation but relation may not be a
  function.

Relation
Function
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Question Given a graph, how do you know it is
the graph of a function or not ?
Vertical Line Test for Graph of a Function A
graph is the graph of a function if and only if
there is no vertical line crosses the graph more
than once.
Ex.
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Inverse Functions
Inverse function seeks the reverse mapping from
the output to input

• Given 2 functions, f(x) g(x), if f(g(x)) x
  AND g(f(x)) x, then f(x) g(x) are inverses of
  each other.

Symbols f -1(x) means f inverse of x
f -1(x) g(x), g-1(x) f(x)
f(f -1(x)) x, f -1(f(x)) x
f -1
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Question Given a function, how do you know
there exists its inverse function or not ?
A function is one-to-one if each output in the
range is mapped from one, and only one input, x,
in the domain.
Horizontal Line Test If each horizontal line
crosses the graph of a function at no more than
one point, then the function is a one-to-one
function.
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Example The function y x2 4x 7 is not
one-to-one on the real numbers because the line y
7 intersects the graph at both (0, 7) and (4,
7).
(4, 7)
(0, 7)
y 7
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Example Apply the horizontal line test to the
graphs below to determine if the functions are
one-to-one.
a) y x3
b) y x3 3x2 x 1
one-to-one
not one-to-one
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Horizontal Line Test

• Used to determine whether a functions inverse
  will be a function by seeing if the original
  function passes the horizontal line test.
• If the original function passes the horizontal
  line test, then its inverse is a function.
• If the original function does not pass the
  horizontal line test, then its inverse is not a
  function, or, the inverse does not exist.

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To find the inverse of a function

• Solve for x.
• Switch the y x values.
• State the new domain(old range) and new range(old
  domain).
• Remember functions have to pass the vertical
  line test!

Relation
Function
Inverse Function
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Ex Find an inverse of y 3x6.

• Steps - solve for x - switch x y
• y 3x6
• 3x y 6

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The inverse function is an inverse with respect
to the operation of composition of functions.
The inverse function undoes the function, that
is, f -1( f (x)) x.
The function is the inverse of its inverse
function, that is, f ( f -1(x)) x.
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Example Verify that the function g(x)
is the inverse of f(x) 2x 1.
It follows that g f -1.
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Ex Verify that f(x)-3x6 and g(x)-1/3x2 are
inverses.

• Meaning find f(g(x)) and g(f(x)). If they both
  equal x, then they are inverses.

f(g(x)) -3(-1/3x2)6 x-66 x
g(f(x)) -1/3(-3x6)2 x-22 x
Because f(g(x))x and g(f(x))x, they are
inverses.
f -1(x) g(x) and g-1(x) f(x)
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Ex (a) Find the inverse of f(x)x5.
(b) Is f -1(x) a function? (hint look at the
graph! Does it pass the vertical line test?)

1. y x5

Yes , f -1(x) is a function.
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Ex Graph the function f(x)x2 and determine
whether its inverse is a function.
Graph does not pass the horizontal line test,
therefore the inverse is not a function.
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Ex f(x)2x2-4 Determine whether f -1(x) is a
function, then find the inverse equation.
y 2x2-4 y 4 2x2
f -1(x) is not a function.
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Ex g(x) 2x3
y 2x3
OR, if you fix the tent in the basement
Inverse is a function!
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The graphs of a relation and its inverse are
reflections in the line y x.
y x
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Example From the graph of the function y f
(x), determine if the inverse relation is a
function and, if it is, sketch its graph.
y
y f -1(x)
y x
The graph of f passes the horizontal line test.
y f(x)
x
The inverse relation is a function.
Reflect the graph of f in the line y x to
produce the graph of f -1.
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Assignment
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A function is a mapping from its domain to its
range so that each element, x, of the domain is
mapped to one, and only one, element, f (x), of
the range.
A function is one-to-one if each element f (x) of
the range is mapped from one, and only one,
element, x, of the domain.