We're asked to select all of the statements
that are true given the piecewise defined function f of x.
We're given f of x equals 2x minus three if x is less than
three, f of x equals the square root of the quantity
x minus two if x is greater than three, and f of x
equals negative four if x equals three.
Now we will determine which of the given statements
are true.
We'll first determine this analytically and then verify
the results graphically the first statement is f of three
is defined.
Going back to the piecewise defined function we are given
f of x equals negative four if x equals three, which means
f of three equals negative four and therefore f of three
is defined, we select this first statement
because it is true.
Next the limit as x approaches three of f of x exists,
remember if this limit exists, we must approach the same
function value from the left and right of positive three,
which means the limit as x approaches three
from the negative side or left side of f of x must equal
the limit as x approaches three from the positive side
or right side of f of x.
Let's determine both one set of limits and see if we get
the same value.
As we approach x equals three from the negative side
or left side we are approaching x equals three from values
that are less than three which means we are approaching
along the function f of x equals 2x minus three because this
is the function rule if x is less than three.
And we know we can determine a limit of a linear function
by performing direct substitution and therefore substituting
three into the linear function gives us two times three
minus three, six minus three is equal to three.
And now for this limit to exist we must also get three
for the limit as x approaches three from the right
or positive side.
So as we approach x equals three from the positive or right
side we are approaching three from values that are greater
than three and therefore we are approaching along the curve
given by f of x equals the square root of the quantity
x minus two because this is the function rule if x is
greater than three.
And as along as the value we are approaching is in
the domain of the square root function we can determine
this limit by performing direct substitution.
So if we substitute three into the square root function,
we have the square root of the quantity three minus two
which equals the square root of one which equals one.
Notice how the one set of limits are not equals and
therefore the limit as x approaches three of f of x does not
exist, the statement is not true.
The next statement is the limit as x three of f of x
equals f of three which obviously cannot be true
because the limit doesn't exist.
The statement applies we can determine this limit by
performing direct substitution which we know is not true.
The next statement is the function is continuous
at x equals three.
Using the definition of continuity at a point, all three
of these statements must be true in order for the function
to be continuous at a particular x value.
And therefore this statement is not true and the last
statement is the function is not continuous at x equals
three which is true.
Again let's verify these results graphically.
Notice how for the graph of the piecewise defined function,
we have the line f of x equals 2x minus three if x is less
than three, which is this piece of the graph.
Notice how we have a hole here at x equals three because
when x equals three, the function value is negative four
which gives us the closed point down here.
And then when x is greater than three, we have the square
root function f of x equals the square root of the quantity
x minus two which is this piece of the graph here.
So going back to the statements, f of three is defined,
f of three is defined as true, because we have a closed
point here where x equals three.
The next statement is the limit as x approaches three
of f of x exists, which we know is not true, because as we
approach three from the left or negative side, notice how
we are approaching the function value of three and from
the right or positive side, we are approaching the function
value of positive one.
The limit does not exist, the statement is not true.
Next if the limit as x approaches three of f of x equals
f of three but is not true because a limit doesn't exist.
The function is continuous at x equals three, graphically
we can see there's a break in the graph and therefore
the function is not continuous, which is the reason why
the last statement is true.
I hope you found this helpful.