Inverse Functions and Relations

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Inverse Functions and Relations
By John Michael Ikard
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What You Will Learn

• Find the inverse of a function or relation
• Determine whether two functions or relations are
  inverses

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How Are Inverse Functions Related To Measurement
Conversions?
The United States uses customary measurements
such as miles per hour. Most scientific formulas
involve measurements given in SI (international
system) units. To make the conversion from miles
per hour to meters per second, simple steps must
be taken. f(x) x miles . 1600 meters . 1
hour or f(x) 4/9x. To 1 hour 1
mile 3600 sec
convert x meters per second to an
approximate equivalent in miles per hour, you can
evaluate g(x) x meters . 3600 seconds . 1 mile
or g(x) 9/4x. These 1 sec 1 hour
1600 miles functions are inverses of one
another
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Vocabulary
INVERSE RELATION the set of ordered pairs
obtained by reversing the coordinates of each
original ordered pair INVERSE FUNCTION two
functions f and g are inverse functions if and
only if both of their compositions are the
identity function IDENTITY FUNCTION the function
I(x) x ONE-TO-ONE a function where each
element of the range is paired with exactly one
element of the domain.
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Finding An Inverse Relation

• KEY CONCEPT Two relations are inverse relations
  if and only if whenever one relation contains the
  element (a,b) the other contains the element
  (b,a).
• EXAMPLE Q (1,2) (4,9) (3,5)
• S (2,1) (9,4) (5,3)
• Q and S are inverse relations
• This is prevalent in geometry when trying to
  find the reflection of a specific shape on a
  grid.

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Find Inverse Function
Follow along on page 391 example 2. Example
will be worked out KEY CONCEPT Suppose f and f
1 are inverse functions. Then, f(a) b if and
only if f 1 (b) a. Step 1 Replace f(x) with y
in the original problem Step 2 Interchange x
and y Step 3 Solve for y Step 4 Replace y with
f 1 (x).
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Verifying Two Functions Are Inverses
Determine whether f(x) 5x10 and g(x) 1/5x
2 are inverse functions (f o g) f(g(x) ) (g o
f)(x) g(f(x) ) f( 1/5x 2) g (5x
10) 5( 1/5x 2) 10
1/5 (5x 10) - 2 x 10 10 x 2
2 x x The functions are
inverses since both (f o g) (x) and (g o f) (x)
equal x.
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DONE