3.4 Use Inverse Functions

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3.4 Use Inverse Functions

• p. 190
• What is an inverse relation?
• What do you switch to find an inverse relation?
• What notation is used for an inverse function?
• How is it read?
• What test can you use to verify the inverse of a
  function is a function?

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Review

• 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

x y2
the inverse of an equation switch the x y
and solve for 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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Ex Find an inverse of y -3x6.

• Steps -switch x y
• -solve for y
• y -3x6
• x -3y6
• x-6 -3y

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Find an equation for the inverse of the relation
y 3x 5.
y 3x 5
Write original relation.
Switch x and y.
x 3y 5
Add 5 to each side.
x 5 3y
Solve for y. This is the inverse relation.
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Inverse Functions

• 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
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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.
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To find the inverse of a function

• Change the f(x) to a y.
• Switch the x y values.
• Solve the new equation for y.

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SOLUTION
STEP 1
STEP 2
Show that f(f 1(x)) x.
Show that f 1(f(x)) x.
x 5 5
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SOLUTION
STEP 1
Find the inverse function.
Write original model.
Add 5 to each side.
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Evaluate the inverse function when R 19.
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Find the inverse of f(x) x2, x 0. Then graph
f and f 1.
SOLUTION
f(x) x2
Write original function.
y x2
Replace f (x) with y.
x y2
Switch x and y.
Take square roots of each side.
The domain of f is restricted to nonnegative
values of x. So, the range of f 1 must also be
restricted to nonnegative values, and therefore
the inverse is f 1(x) x. (If the domain was
restricted to x 0, you would choose f 1(x)
x.)
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Vertical Line Test

• A vertical line test for functions can be used to
  see if the relation is a function.
• A relation is a function if and only if no
  vertical line intersects the graph of the
  relation at more than one point.

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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
2. x y5

Yes , f -1(x) is a function.
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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.

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Ex g(x)2x3
y2x3 x2y3
OR, if you fix the tent in the basement
Inverse is a function!
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Consider the function f (x) 2x3 1. Determine
whether the inverse of f is a function. Then find
the inverse.
SOLUTION
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Find the inverse of a cubic funtion
f (x) 2x3 1
Write original function.
y 2x3 1
Replace f (x) with y.
Switch x and y.
x 2y3 1
x 1 2y3
Subtract 1 from each side.
Divide each side by 2.
Take cube root of each side.
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Find the inverse of the function. Then graph the
function and its inverse.
5. f(x) x6, x 0
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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 x 2y2-4 x4 2y2
OR, if you fix the tent in the basement
f -1(x) is not a function.
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• What is an inverse relation?
• It maps the output values back to the original
  input values (domain or inverse is range or
  original).
• What do you switch to find an inverse relation?
• x and y
• What notation is used for an inverse function?
• f -1 (x)
• How is it read?
• f inverse of x
• What test can you use to verify the inverse of a
  function is a function?
• Horizontal line test

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Assignment
Page194, 3-39 every 3rd problem, 46, 47