Download PPT : Inverse Functions

• Given any function, f, the inverse
of the function, f -1, is a relation
that is formed by interchanging
each (x, y) of f to a (y, x) of f -1.
Let f be defined as the set of values given by
x-values
-2
0
4
7
y-values
0
4
-5
10
Let f -1 be defined as the set of values given by
x-values
0
4
-5
10
y-values
-2
0
4
7
Function 1
6
x
-2
y
13
0
7
4
-5
y
4
2
x
7
-14
–6
Function 2
x
13
y
-2
7
0
-5
-14
4
7
–4
–2
2
–2
–4
–6
y=x
4
6
Let
x
y
5x  3
y
x
5y  3
To find the inverse, switch x and y,
Solve for y: x(5 y  3)  y
5 xy  3x  y
5 xy  y  3x
y5x 1  3x
3x
y
5x  1
So the inverse of
f ( x) :
x
y
5x  3
is
3x
f ( x) : y 
5x 1
1
y  2x  4
x  2y  4
2y  x  4
1
y  x2
2
1. Exchange x and y
2. Solve for y.
3. Graph both lines.
4. Graph
yx
5. What does this
line represent?
6
y
4
Equation 2: y=.5x+2
2
x
−6
−4
−2
2
−2
−4
−6
4
Equation 1: y=2x−4
6
y  2x  4
2
x  2y  4
2
2y  x  4
1
2
y  x2
2
1. Exchange x and y
2
2. Solve for y.
3. Graph both curves.
yx
4. Graph
5. What does this
line represent?
6
y
4
2
x
1
y
x2
2
−10
−5
5
−2
−4
−6
y2
y
x2
1. Exchange x and y
x
2. Solve for y.
In this case y is the exponent. How could we solve for y.
Mathematicians had to come up with a new term to
represent the solution of this equation.
6
y2
y  log 2 x
x
y
4
2
x
−10
−5
5
−2
−4
−6
y  log 2 x
Rewrite the following Exponential Equations into Logarithmic
Equations
EXAMPLE 1
2 8
3
Base
Power
log 2 8  3
Exponent
Base
(Argument)
Exponent
Power
(Argument)
Rewrite the following Exponential Equations into Logarithmic
Equations
EXAMPLE 2
1
10 
1000
3
Base
Power
1
log 10
 3
1000
Exponent
Base
(Argument)
Exponent
Power
(Argument)
Rewrite the following Logarithmic Equations into Exponential
Equations
EXAMPLE 3
log 2 32  5
2  32
5
Base
Base
Exponent
Exponent
Power
Power
(Argument)
(Argument)
Rewrite the following Logarithmic Equations into Exponential
Equations
EXAMPLE 4
log 3 27  3
3  27
3
Base
Base
Exponent
Exponent
Power
Power
(Argument)
(Argument)