An inverse is the
that takes you back to where you started. The inverse of
is division, adding and subtracting, square and square root, etc.
are two conditions for a function to be the inverse function:
1) g(f(x)) = x for all x in the domain of f
2) f(g(x)) = x for all x in the domain of g
Notice in both
you will get back to the element that you started with, namely, x.
The notation used
indicate an inverse function is: f -1(x)
pronounced "f inverse". This
notation does not mean 1/f(x).
1) If f(x) =
- 1 and g(x) = (x + 1)/3, show that f and g are inverses to each other.
To show that they are inverses, we must prove both of the above parts.
g(f(x)) = g(3x - 1) = (3x - 1 + 1)/3 = 3x/3 = x
f(g(x)) = f((x + 1)/3) = 3[(x + 1)/3] - 1 = x + 1 - 1 = x
Since both parts
they are indeed inverses of each other.
To find a rule
the inverse function
Find the inverse
for y = 5x + 2
To find the
interchange x and y.
x = 5y + 2
Now isolate for y!!
x - 2 = 5y
(x - 2)/5 = y
We now have the
Notice, that this
make sense. The original problem had adding by two and the
is subtracting two. The original function had multiplying by five
and the inverse has division by five.
We have to make
that the inverse is indeed a function. Not all functions will
inverses that are also functions. In order for a function to have
an inverse, it must pass the horizontal line test!!
Horizontal line test
If the graph of a
y = f(x) is such that no horizontal line intersects the graph in more
one point, then f has an inverse function.
This will make
when we discover how to graph the inverse function. To graph the
inverse function, it is simply the reflection about the line y =
Makes sense, because in order to get the graph, we interchange x and
Recall from previous sections what the reflection about the line y = x
looks like. Any two points on the same horizontal line when
will be on the same vertical line. Can't have this because it
be a function. That's why the horizontal line test works.
1) Find the
of f -1
graph f, f -1,
y = x for f(x) = 2x - 5.
First, f(x) is a line and it passes the horizontal line test.
Find the inverse: y = 2x - 5
x = 2y - 5
x + 5 = 2y
(x + 5)/2 = y
2) Let f(x) = 9 - x2
for x > 0
Find the equation for f -1(x)
Sketch the graph of f, f -1,
and y = x.
Notice that the equation is half of a parabola. Only the side to
the right of zero. If we tried to use the entire parabola, it
pass the horizontal line test.
To find the equation: y = 9 - x2
x = 9 - y2
x - 9 = -y2
9 - x
x < 9
On to the last
Back up and regroup: