6'3 Inverse Functions

1
6.3 Inverse Functions

• To find an inverse of a function, switch the
  coordinates of the ordered pairs.
• Example
• Relation (0,-3), (1,-1), (2,1)
• Inverse (-3,0), (-1,1), (1,2)
• Therefore, an inverse is a reflection of the
  original relation.
• The line of reflection is the line y x, a 45
  line drawn through the origin of the x-y plane.

2
Example

• Find the inverse of f(x) 3x 1
• Step 1 Replace f(x) with y.
• y 3x 1 (graphing m 3/1, y-intercept
  (0,-1)
• Step 2 Switch x and y.
• x 3y 1
• Step 3 Write equation in slope intercept form.
• 3y 1 x
• 3y 1 1 x 1
• 3y x 1
• 3y/3 x/3 1/3
• y (1/3)x 1/3 (graphing m 1/3,
  y-intercept (0,1/3)

3
Graphing

• Given y 3x 1, we found the inverse, y (1/3)x
  1/3
• Graphically, we would have two lines.
• Each line would be a reflection of the other over
  the line of reflection.

4
Verifying Inverses

• Rule If f(g(x)) x and g(f(x)) x then f and g
  are inverses.
• Example
• Using the previous example f(x) 3x 1 and g(x)
  (1/3)x 1/3, check algebraically if these
  functions are inverses.
• Solution
• f(g(x)) 3 (1/3 x 1/3) 1 Replaced x with
  g(x).
• x 1 1 x
• g(f(x) 1/3 (3x 1) 1/3 Replaced x with
  f(x).
• x 1/3 1/3 x
• Since f(g(x)) x and g(f(x) x the functions
  are inverses.