MTH-112 Quiz 1
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
3. Does the equation x2 y − 1 = 63 define y as a
function of x?
1. Determine whether the given relation is a function. Give the domain and range of the relation.
{(1, 2), (2, 1), (3, 1), (4, 1)}
(a) Is this a function?
(b) What is the domain of the relation?
4. In the graph is y a function of x? (yes / no)
y
(c) What is the range of the relation?
6
5
4
3
2
2. Evaluate the function f (x) = −3x4 − 4x3 − 13
at the following values.
1
-4 -3 -2 -1
-1
(a) f (−3)
1
2
3
4
3
4
x
-2
5. Use the graph to find g(−2).
g(−2) =
(b) f (2a)
y
3
2
1
-5 -4 -3 -2 -1
-1
g(x)
(c) f (−x)
1
2
5
x
-2
-3
6. Find the domain and range of the above function g(x). Write your answers in interval notation.
Domain =
Range =
1
MTH-112 Quiz 1 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
4. In the graph is y a function of x? ( yes / no)
1. Determine whether the given relation is a function. Give the domain and range of the relation.
If any vertical line crosses the graph at more than
one point, then the graph is not a function.
{(1, 2), (2, 1), (3, 1), (4, 1)}
If every vertical line crosses the graph only once,
then the graph is a function.
(a) Is this a function?
Yes. The first components do not repeat.
(b) What is the domain of the relation?
{1, 2, 3, 4} The first components of ordered
pairs. When listing components use braces
{ }, not parentheses ( ).
y
6
5
4
(c) What is the range of the relation?
{2, 1} The second components of ordered
pairs.
3
2
1
2. Evaluate the function f (x) = −3x4 − 4x3 − 13
at the following values.
-4 -3 -2 -1
-1
2
3
4
x
-2
(a) To find f (−3), replace x with −3.
f (−3) = −3(−3)4 − 4(−3)3 − 13
5. Use the graph to find g(−2).
= −3(81) − 4(−27) − 13
= −148
g(−2) means the y− coordinate of the point on
the graph, for which the x− coordinate is −2.
(b) To find f (2a), replace x with 2a.
f (2a) = −3(2a)4 − 4(2a)3 − 13
4
1
g(−2) = −1
3
= −3(16a ) − 4(8a ) − 13
y
= −48a4 − 32a3 − 13
3
(c) To find f (−x), replace x with −x.
4
2
1
3
f (−x) = −3(−x) − 4(−x) − 13
= −3x4 + 4x3 − 13
-5 -4 -3 -2 -1
-1
3. Does the equation x2 y − 1 = 63 define y as a
function of x?
g(x)
1
2
3
4
5
x
-2
-3
Solve the equation for y in terms of x.
6. Find the domain and range of the above function g(x). Write your answers in interval notation.
x2 y = 64
64
y= 2
x
Domain is the set of x− coordinates of the points
on the graph.
Pick an x value; substitute, and find y. If there is
only one y value, the equation is a function.
64
64
Let x = 3: Then y = 2 =
x
9
There is only one y−value.
Domain = [−4, 4)
Range is the set of y− coordinates of the points
on the graph.
Range = [−2, 2)
Yes.
1
MTH-112 Quiz 2
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
1. Use the graph to find the following:
2. Find the function value of given x values. Then
graph the function.
y
5
f (x) =
4
f (x)
−3x + 6
−2
if x ≤ 2
if x > 2
3
(a) f (1) =
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
x
-2
(b) f (2) =
-3
-4
-5
(a) In which interval, if any, is the function
increasing?
(c) f (3) =
(b) In which interval, if any, is the function
decreasing?
(d) f (4) =
(c) In which interval, if any, is the function a
constant?
(d) The relative minimum is (write your answer as an ordered pair):
(e) Graph the function.
y
(e) The relative maximum is (write your answer as an ordered pair):
6
5
4
3
(f) Domain:
2
1
-6 -5 -4 -3 -2 -1
-1
(g) Range:
-2
-3
(h) Is the function graphed even, odd or neither? (even / odd / neither)
-4
-5
-6
1
1
2
3
4
5
6
x
3. (2, −3) is a point on the graph of an even function. What other point must be on the graph?
(Write your answer as an ordered pair.)
6. It is recommended that, for every hour you
spend in class, you study two hours on your
own. (That is, if you spend three hours in class,
you study six hours on your own, and so on.)
Let x represent the number of hours you spend
in class, and y represent the recommended number of hours of study on your own. Write y as
a function of x, in the equation form.
4. Determine whether the function is even, odd or
neither.
(a) f (x) = x4 + 3x + 7 (even / odd / neither)
7. The graph of a function f (x) is given below.
Graph g(x) = f (x − 1) − 2 on the same coordinate plane.
(b) f (x) = x3 + 4x + 7 (even / odd / neither)
y
6
5
4
f (x)
3
2
5. For the function f (x) = −4x + 44, find:
1
(a) f (x + h)
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
x
-2
-3
-4
-5
-6
(b) The difference quotient:
f (x + h) − f (x)
h
8. Is the given function f (x) in problem 7 even,
odd or neither? (even / odd / neither)
2
MTH-112 Quiz 2 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
A relative minimum is a point on the graph
where the graph changes from decreasing to
increasing.
1. Use the graph to find the following:
y
5
(−3, −2)
4
f (x)
(e) The relative maximum is (write your answer as an ordered pair):
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
A relative maximum is a point on the graph
where the graph changes from increasing to
decreasing.
x
-2
(1, 2)
-3
-4
(f) Domain:
-5
Domain is the set of x− coordinates of the
points on the graph.
(a) In which interval, if any, is the function
increasing?
Domain = (−∞, ∞)
From left to right, if the graph goes up, it
is increasing. When writing increasing, decreasing, and constant intervals, use x values
(not y values). In the graph, the increasing
part is in blue color, and the corresponding
interval on the x- axis is shaded in blue color.
(g) Range:
(−3, 1)
(h) Is the function graphed even, odd or neither? (even / odd / neither )
Range is the set of y− coordinates of the
points on the graph.
Range = (−∞, ∞)
(b) In which interval, if any, is the function
decreasing?
y
From left to right, if the graph goes down, it
is decreasing. In the graph, the decreasing
parts are in red color, and the corresponding intervals on the x- axis are shaded in red
color.
5
4
(−1, 2)
3
(1, 2)
2
1
(−∞, −3) ∪ (1, ∞)
-6 -5 -4 -3 -2 -1
-1
(c) In which interval, if any, is the function a
constant?
-2
(−1, −2) -3
From left to right, if the graph does not go
up or down, it is constant.
1
2
3
4
5
6
f (x)
-4
-5
None.
The point (1, 2) is on the graph. But the
symmetric point about the y−axis, which is
(−1, 2), is not on the graph. Therefore the
(d) The relative minimum is (write your answer as an ordered pair):
1
x
graph is not symmetric with respect to the
y−axis. Thus, the function is not even.
y
6
The point (1, 2) is on the graph. But the
symmetric point about the origin, which is
(−1, −2), is not on the graph. Therefore
the graph is not symmetric with respect to
the origin. Thus, the function is not odd.
5
4
3
2
1
2. Find the function value of given x values. Then
graph the function.
−3x + 6 if x ≤ 2
f (x) =
−2
if x > 2
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
-2
-3
-4
(a) Since the x−value 1 is less than 2, use the
top part of the function to find f (1), that is,
f (x) = −3x + 6.
-5
-6
3. (2, −3) is a point on the graph of an even function. What other point must be on the graph?
(Write your answer as an ordered pair.)
f (1) = −3(1) + 6 = 3
(b) Since the x−value 2 is equal to 2, use the
top part of the function to find f (2), that is,
f (x) = −3x + 6.
The symmetric point of (2, −3) with respect to
y−axis must also be on the graph.
f (2) = −3(2) + 6 = 0
(−2, −3)
(c) Since the x−value 3 is greater than 2, use
the bottom part of the function to find f (3),
that is, f (x) = −2.
4. Determine whether the function is even, odd or
neither.
f (3) = −2
(a) f (x) = x4 + 3x + 7 (even / odd / neither )
(d) Since the x−value 4 is greater than 2, use
the bottom part of the function to find f (4),
that is, f (x) = −2.
Since there are even exponents (4 and 0),
and an odd exponent (1), the function is
neither even nor odd.
f (4) = −2
(The constant 7 is the same as 7x0 , and the
exponent 0 is an even number.)
(e) Graph the function.
Separate the coordinate plane into two
parts using the vertical line x = 2;
plot the four points found above
(1, 3), (2, 0), (3, −2), (4, −2); then draw the
graphs through those points.
(b) f (x) = x3 + 4x + 7 (even / odd / neither )
Since there are odd exponents (3 and 1),
and an even exponent (0), the function is
neither even nor odd.
The right end point of the left piece must be
a closed circle (because the left piece represents the graph for x ≤ 2).
5. For the function f (x) = −4x + 44, find:
(a) To find f (x + h), replace x with x + h.
The left end point of the right piece must
be on the vertical dotted line, and an open
circle (because the right piece represents the
graph for x > 2).
f (x + h) = −4(x + h) + 44
= −4x − 4h + 44
2
x
(b) The difference quotient:
f (x + h) − f (x)
h
−1 shifts the graph 1 unit to the right, and −2
shifts the graph 2 units down.
−4x − 4h + 44 − (−4x + 44)
h
−4x − 4h + 44 + 4x − 44
=
h
−4h
=
h
= −4
=
y
6
5
4
f (x)
3
g(x)
2
6. It is recommended that, for every hour you
spend in class, you study two hours on your
own. (That is, if you spend three hours in class,
you study six hours on your own, and so on.)
Let x represent the number of hours you spend
in class, and y represent the recommended number of hours of study on your own. Write y as
a function of x, in the equation form.
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
x
-2
-3
-4
-5
-6
x : the number of hours in class
y : the recommended number of hours of study
on your own
8. Is the given function f (x) in problem 7 even,
odd or neither? ( even / odd / neither)
y = 2x
The point (2, 4) is on the graph of f (x). The symmetric point about the y−axis, which is (−2, 4),
is also on the graph. Therefore the graph f (x) is
symmetric with respect to the y−axis. Thus, the
function f (x) is even.
7. The graph of a function f (x) is given below.
Graph g(x) = f (x − 1) − 2 on the same coordinate plane.
3
MTH-112 Quiz 3
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
1. Use the graph to find the following:
2. In the graph is y a function of x? (yes / no)
y
y
6
f (x)
4
5
3
4
2
3
1
2
-6 -5 -4 -3 -2 -1
-1
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
x
1
2
3
4
5
6
x
-2
-2
-3
3. For the function f (x) = 3x2 + 5x − 9, find:
-4
-5
(a) f (x + h)
-6
(a) In which interval, if any, is the function
increasing?
(b) In which interval, if any, is the function
decreasing?
(c) In which interval, if any, is the function a
constant?
(b) The difference quotient:
(d) Domain:
(e) Range:
(f) Is the function graphed even, odd or neither? (even / odd / neither)
(g) Graph g(x) = −f (x − 2) on the same coordinate plane.
1
f (x + h) − f (x)
h
4. Graph the function.
2x − 1
f (x) =
−x + 4
6. Find the domain of the following functions.
(Write your answers as intervals.)
if x < 1
if x ≥ 1
(a) f (x) =
x2
x−1
− x − 12
y
6
5
4
3
Domain:
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
x
(b) g(x) =
√
−x − 10
-2
-3
-4
-5
-6
5. Determine whether the function is even, odd or
neither.
Domain:
(a) f (x) = x4 + 2x2 + 5 (even / odd / neither)
(c) h(x) = x5 + 3x3 + 5
(b) f (x) = x3 + 2x + 5 (even / odd / neither)
Domain:
2
MTH-112 Quiz 3 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
(d) Domain:
1. Use the graph to find the following:
Domain is the set of x− coordinates of the
points on the graph.
y
6
f (x)
Domain: (−∞, ∞)
5
4
(e) Range:
3
Range is the set of y− coordinates of the
points on the graph.
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
Range: [1, ∞)
x
(f) Is the function graphed even, odd or neither? (even / odd / neither )
-2
-3
g(x)
y
-4
6
-5
-6
5
f (x)
(−2, 3) 4
3
(a) In which interval, if any, is the function
increasing?
(2, 3)
2
From left to right, if the graph goes up, it
is increasing. When writing increasing, decreasing, and constant intervals, use x values
(not y values). In the graph, the increasing
part is in blue color, and the corresponding
interval on the x- axis is shaded in blue color.
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
-2
(−2, −3)
-3
-4
(0, ∞)
-5
(b) In which interval, if any, is the function
decreasing?
-6
The point (2, 3) is on the graph. But the
symmetric point about the y−axis, which is
(−2, 3), is not on the graph. Therefore the
graph is not symmetric with respect to the
y−axis. Thus, the function is not even.
From left to right, if the graph goes down, it
is decreasing. In the graph, the decreasing
part is in red color, and the corresponding
interval on the x- axis is shaded in red color.
(−∞, −2)
The point (2, 3) is on the graph. But the
symmetric point about the origin, which is
(−2, −3), is not on the graph. Therefore
the graph is not symmetric with respect to
the origin. Thus, the function is not odd.
(c) In which interval, if any, is the function a
constant?
From left to right, if the graph does not go
up or down, it is constant. In the graph, the
constant part is in green color, and the corresponding interval on the x- axis is shaded
in green color.
(g) Graph g(x) = −f (x − 2) on the same coordinate plane.
−2 shifts the graph 2 units to the right, and
− sign at the beginning reflects the graph
over the x- axis.
(−2, 0)
1
x
Separate the coordinate plane into two parts using
the vertical line x = 1.
2. In the graph is y a function of x? ( yes / no)
If any vertical line crosses the graph at more than
one point, then the graph is not a function.
Find two points on each side of the vertical line
(find the y− values for any two x− values less
than 1, and any two x− values greater than or
equal to 1).
If every vertical line crosses the graph only once,
then the graph is a function.
f (−1) = 2(−1) − 1 = −3
y
4
f (0) = 2(0) − 1 = −1
3
f (1) = −(1) + 4 = 3
2
f (2) = −(2) + 4 = 2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
Plot
the
four
points
found
above
(−1, −3), (0, −1), (1, 3), (2, 2); then draw the
graphs through those points.
x
-2
The right end point of the left piece must be
on the vertical dotted line, and an open circle
(because the left piece represents the graph for
x < 1).
3. For the function f (x) = 3x2 + 5x − 9, find:
(a) f (x + h)
The left end point of the right piece must be a
closed circle (because the right piece represents
the graph for x ≥ 1).
To find f (x + h), replace x with x + h.
f (x + h)
= 3(x + h)2 + 5(x + h) − 9
y
= 3(x2 + 2hx + h2 ) + 5x + 5h − 9
6
= 3x2 + 6hx + 3h2 + 5x + 5h − 9
5
4
(b) The difference quotient:
f (x + h) − f (x)
h
3
2
1
f (x + h) − f (x)
h
3x2 + 6hx + 3h2 + 5x + 5h − 9 − (3x2 + 5x − 9)
=
h
3x2 + 6hx + 3h2 + 5x + 5h − 9 − 3x2 − 5x + 9)
=
h
6hx + 3h2 + 5h
=
h
h(6x + 3h + 5)
=
h
= 6x + 3h + 5
4. Graph the function.
2x − 1
f (x) =
−x + 4
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
x
-2
-3
-4
-5
-6
5. Determine whether the function is even, odd or
neither.
(a) f (x) = x4 +2x2 +5 ( even / odd / neither)
Since all the exponents (4, 2 and 0) are
even, the function is even.
if x < 1
if x ≥ 1
2
(b) f (x) = x3 + 2x + 5 (even / odd / neither )
The domain is all real numbers except 4 and
−3.
Domain: (−∞, −3) ∪ (−3, 4) ∪ (4, ∞)
√
(b) g(x) = −x − 10
Since there are odd exponents (3 and 1),
and an even exponent (0), the function is
neither even nor odd.
6. Find the domain of the following functions.
(Write your answers as intervals.)
(a) f (x) =
Inside the square root must be non-negative.
−x − 10 ≥ 0
x−1
2
x − x − 12
−10 ≥ x
The domain is all real numbers except the
zeros of the bottom polynomial.
Domain: (−∞, −10]
(c) h(x) = x5 + 3x3 + 5
x2 − x − 12 = 0
Domain of any polynomial is all real numbers.
(x − 4)(x + 3) = 0
x=4
x = −3
Domain: (−∞, ∞)
3