Domain/Range, Vertical Line Test, Increasing/Decreasing/Constant Functions, Even/Odd Functions, and Greatest Integer Function

We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look to the left and
right and see
if there are any end points. In

this case, note how there are arrows on both ends of the graph and
no end points. This means that the graph goes on and on forever
in
both directions.

**This means that the domain is .**

We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look up and down and
see if there
are any end points. In this case, note how the graph has a upper
endpoint of *y* = -3 and it has arrows
going
down from that.

**This means that the range is .**

If the

If you said there is none, you are right.

Since the graph never crosses the *x*-axis,
then **there is no x-intercept.**

If the

If you said *y* = -3 you
are correct.

**The ordered pair for this y-intercept
would be (0, -3).**

If the functional value correlates with the second or

If you said *f*(-2) = -5 ,
then give yourself
a pat on the back. The functional value at *x* = -2 is -5.

**The ordered pair for this would be (-2, -5).**

We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how there are no endpoints and the graph goes on and on forever in both directions.

**This means that the domain is **.

We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look up and down and see if there are any end points. In this case, note how there are no endpoints and the graph goes on and on forever in both directions.

**This means that the range is .**

If the

If you said *x* = 1
you are correct.

**The ordered pair for this x-intercept
would be (1, 0).**

If the

If you said *y* = -2 you
are correct.

**The ordered pair for this y-intercept
would be (0, -2).**

If the functional value correlates with the second or

If you said *f*(6) = 2 ,
then give yourself
a pat on the back. The functional value at *x* = 6 is 2.

**The ordered pair for this would be (6, 2).**

This graph would not pass the vertical line test
because there is at
least one place on it that we could draw a vertical line and intersect
it in more than one place. In fact, there are a lot of vertical
lines
that we can draw that would intersect it in more than one place, but we
only need to show one to say it is not a function.

The graph below shows one vertical line drawn through our graph that intersects it in two places: (1, 2) and (1, -2). This shows that the input value of 1 associates with two output values, which is not acceptable in the function world.

**Therefore, this is not a graph of a function.**

This graph would pass the vertical line test, because
there would not
be any place on it that we could draw a vertical line and it would
intersect
it in more than one place.

**Therefore, this is a graph of a function.**

A function is increasing in an interval when it is going up left to right in that interval? With that in mind, what interval, if any, is this function increasing?

**If you said ,
you are correct. **

Note how the function is going up left to right, from
negative infinity
to *x* = 0.

**Below shows the part of the graph that is increasing:**

A function is decreasing in an interval when it is going down left to right in that interval? With that in mind, what interval, if any, is this function decreasing?

**If you said ,
you are right on. **

Note how the function is going down left to right
starting at *x *=
0 and everywhere to the right of that.

**Below shows the part of the graph that is decreasing:**

A function is constant in an interval if it is horizontal in the entire interval. With that in mind, what interval, if any, is this function constant?

**If you said it is never constant, pat yourself on the
back. **

Note how the function is never a horizontal line.

A function is increasing in an interval when it is going up left to right in that interval? With that in mind, what interval, if any, is this function increasing?

**If you said or ,
you are correct. **

Note how the function is going up left to right, from
negative infinity
to *x* = 2 and also starting at *x* = 6 and everywhere to the right of that.

**Below shows the part of the graph that is increasing:**

A function is decreasing in an interval when it is going down left to right in that interval? With that in mind, what interval, if any, is this function decreasing?

**If you said that it was never decreasing you are
right. **

The graph never goes down left to right.

A function is constant in an interval if it is horizontal in the entire interval. With that in mind, what interval, if any, is this function constant?

**If you said (2, 6), pat yourself on the back. **

Note how the function is horizontal starting at *x* = 2 all the way to *x* = 6.

**Below shows the part of the graph that is constant:**

Do you know what that means?

**It means that this function is even.**

(return to problem 4a)

Note how the graph is neither symmetric about the *y*-axis
nor the origin.

Do you know what that means?

**It means that this function is neither even nor odd.**

To determine if this function is even, odd, or
neither, we need
to replace *x* with -*x* and compare *f*(*x*)
with *f*(-*x*):

A function is even if for all

If you said no, you are correct. Note how both of their terms have opposite signs, so .

A function is odd if for all

If you said yes, you are
right.

Looking at ,
note how all of the terms of *f*(-*x*)
and -*f*(*x*)
match up, so .

**Final answer: The function is
odd.**

To determine if this function is even, odd, or
neither, we need
to replace *x* with -*x* and compare *f*(*x*)
with *f*(-*x*):

A function is even if for all

If you said yes, you are
correct. Note how
all of the terms of *f*(*x*)
and f(-*x*) match up, so .

**Final answer: The function is
even.**

To determine if this function is even, odd, or
neither, we need
to replace *x* with -*x* and compare *f*(*x*)
with *f*(-*x*):

A function is even if for all

If you said no, you are correct. Note how their second terms have opposite signs, so .

A function is odd if for all

If you said no, you are
right.

Looking at ,
we see that the signs of the first terms of *f*(-*x*)
and -*f*(*x*)
don’t
match, so .

Since we said no for both even and odd, that leaves us
with our answer
to be neither.

**Final answer: The function is neither even nor odd.**

*f*(-9.1)

We need to ask ourselves, what is the greatest integer
that is less
than or equal to -9.1?

If you said -10, you are correct.

Be careful on this one. We are working with a negative number. -9 is not a correct answer because -9 is not less than or equal to -9.1, it is greater than -9.1.

**Final answer: -10**

Last revised on June 18, 2010 by Kim Seward.

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