Inverse

1
Inverse

• Everything has an inverse.
• Visually find inverse by reflecting over the
  line yx

2
Inverse

• Find the inverse by switching the xs and ys of
  the functions points.
• Example
• Function(1,3),(2,4),(1,5),(0,2)
• Inverse (3,1),(4,2),(5,1),(2,0)

3
Inverse

• Definition (f o fof-1)(x)(f o fof-1)(x)x
• Algebraically find inverse by (fog)(x)(gof)(x)x
  
• Example f(x)4x g(x)x/4
• (fog)(x) f(x/4) (gof)(x) g(4x)
• 4(x/4) (4x)/4
• x x x

4
Inverse

• Everything has an inverse but not inverse
  functions.
• You can find out by switching the xs and ys in
  the equation of the function
• It is an inverse function when there is one
  output for one input.
• Change the y to f-1(x)
• Example
• F(x) 4x3-8
• y 4x3-8
• x 4y3-8
• cubert.((x8)/4)y
• f-1cubert.((x8)/4)

5
Inverse

• If there is more than one output for one input,
  it is called an Inverse Relation
• - just keep it y (dont change to f-1(x))
• Example
• F(x) 2x2 3
• y 2x2 3
• x 2y2 3
• sqrert.((x-3)/2) y
• y sqrert.((x-3)/2)

6
Quadratic Functions

• The quadratic equation in standard form is ax2
  bx c. (a cannot equal 0)
• It can open up or down.
• Axis of Symmetry is x -b/ 2a
• Vertex (-b/ 2a, y-value after substitution)
• You get y-intersection by plugging in 0 for x.
• You get x-intersection(s) by plugging in 0 for y.

7
Quadratic Functions

• Example f(x) 4x22x1
• Opens up
• -2/2(4) -1/4
• aos x-1/4
• 3) (-1/4, ¾)
• 4) y-int (0,1)
• 5) x-int (-1,0), (1/2,0)

8
Polynomials

• Definition anxna(n-1)x(n-1) a(n-2)x(n-2)
• The leading coefficient always follows the
  highest degree.
• Domain is all reals.
• Graph is continuous with no holes.
• Find out how graph looks like by looking at the
  coefficient of highest degree.
• -if the coefficient of highest degree is
  negative, graph is negative.
• - maximum number of turns is (degree-1)

9
Polynomials

• Example f(x) x4 3x3 - 9x2 - 23x - 12
• -is positive graph because
• the coefficient of highest
• degree is positive.
• -Has 3 turns because of
• (degree-1).