# uc3rmw_0107e01

0 (0 Likes / 0 Dislikes)

In this example, we are going to test for
the one-to-one property of a function.
We want to determine which of the
following functions are one-to-one.
The functions are as follows.
In part A, we have f of x equal 2x plus 5.
In part B, we have g of x
equals x squared minus 1.
And in part C, we have h of x
equals 2 times the square root of x.
Recall that a one-to-one
function is one in which every y
value corresponds to exactly one x value.
We will determine whether each function
is one-to-one using two methods--
the algebraic method and
the geometric method--
using the horizontal line test.
Let's start with the algebraic method.
We can determine whether a function
is one-to-one algebraically
by assuming that there are two x values
in the domain of the function, x1 and x2,
such that f of x1 equals f of x2.
Starting with f of x1 equal to f
of x2, we have 2 times x1 plus 5
on the left-hand side equals 2 times
x2 plus 5 on the right-hand side.
We'll subtract 5 from both sides, and
we have 2 times x1 equals 2 times x2.
Now we just divide by 2 on both
sides, and we arrive at x1 equals x2.
So we can see that f of x1 equal
f of x2 implies that x1 equals x2,
and therefore, the function is one-to-one.
Now we move on to part B.
Using the same assumption--
that g of x1 is equal to g of x2--
we have x1 squared minus 1
equals x2 squared minus 1.
We'll go ahead and add 1 to both sides.
We have x1 squared equals x2 squared.
Now we take the square root of both
sides, and we have that x1 equals x2 or x1
equals minus x2.
So this means that there are two distinct
numbers for which g of x1 equals g of x2,
and therefore, g of x is
not a one-to-one function.
Now we move on to part C. We apply the same
assumption-- that h of x1 equals h of x2--
and we have that on the left hand
side, 2 times the square root of x1.
That should be equal to--
on the right-hand side--
2 times the square root of x2.
We divide by 2 on both sides,
and we have the square root of x1
equal to the square root of x2.
Finally, we'll take the square of
both sides, and so x1 equals x2.
So h of x1 equal h of x2
implies that x1 equals x2,
and therefore, the function
h of x is one-to-one.
Now I'm going to show you the
geometric method to testing
for the one-to-one property of a function.
In the geometric method, or the horizontal
line test, we first plot the function.
We then apply the horizontal
line test by trying
to see if there is any
horizontal line we can
draw that will intersect the graph of
the function at more than one point.
If there is no horizontal
line that will pass
through the graph of the
function at more than one point,
it is a one-to-one function.
If there is a horizontal line that will pass
through the graph of the function at more
than one point, it is not
a one-to-one function.
So here we see the graph of all the
functions-- f of x, g of x, and h of x.
We see that no horizontal line intersects
the graph of f or h in more than one point.
Therefore, the functions
f and h are one-to-one.
The function g, on the other
hand, is not one-to-one
because we're able to draw a
horizontal line that intersects
the graph at more than one point.
That line is at y equals
3, but we can see that we
could draw several other horizontal
lines that would intersect the graph of g
at more than one point.
It suffices that we only find one horizontal
line that intersects the graph of g
at more than one point to conclude
that the function g is not one-to-one.
And so that's how we're able to determine
whether a function is one-to-one,
both algebraically and geometrically,
using the horizontal line test.