Download Quiz 1 - 4 Solutions

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. When listing components use braces
{ }, not parentheses ( ).
3
2
1
x
-4
2. Evaluate the function f (x) = 3x4 + 4x3 − 134
at the following values.
-3
-2
-1
1
2
3
4
-1
-2
(a) To find f (−3), replace x with −3.
5. Use the graph to find g(2).
f (−3) = 3(−3)4 + 4(−3)3 − 134
= 3(81) + 4(−27) − 134
g(2) means the y− coordinate of the point on the
graph, for which the x− coordinate is 2.
=1
(b) To find f (2a), replace x with 2a.
g(2) = 1
f (2a) = 3(2a)4 + 4(2a)3 − 134
y
= 3(16a4 ) + 4(8a3 ) − 134
3
= 48a4 + 32a3 − 134
2
(c) To find f (−x), replace x with −x.
4
1
3
x
f (−x) = 3(−x) + 4(−x) − 134
4
-5
-4
-3
-2
3
-1
1
2
3
4
5
-1
= 3x − 4x − 134
g(x)
3. Does the equation x2 + y = 64 define y as a
function of 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.
y = 64 − x2
Pick an x value; substitute, and solve for y. If
there is only one y value, the equation defines a
function. If there are more than one y value, the
equation does not define a function.
Domain is the set of x− coordinates of the point
on the graph.
Domain = (−4, 4]
Let x = 3:
2
y = 64 − 3
y = 55
Range is the set of y− coordinates of the point
on the graph.
There is only one y−value.
Range = (−2, 2]
Yes.
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MTH-112 Quiz 2 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
(a) f (x) = x4 + 3x2 + 7 (even / odd / neither)
1. Use the graph to find the following:
Since all the exponents of the variable, 4,
2 and 0, are even numbers, the function is
even. (The constant 7 is the same as 7x0 ,
and the exponent 0 is an even number.)
(b) f (x) = x3 + 2x + 7 (even / odd / neither)
y
6
5
4
3
Since there are odd exponents (3 and 1),
and an even exponent (0), the function is
neither even nor odd.
2
1
.
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
3. Find the function value of given x values. Then
graph the function.
{
4
if x < 2
f (x) =
−3x + 6 if x ≥ 2
-1
-2
For increasing, decreasing or constant intervals,
always give the x−intervals (not y− intervals),
and use parentheses ( ), not brackets [ ].
(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) = 4.
(a) In which interval, if any, is the function
increasing?
f (1) = 4
From left to right, if the graph is going up,
it is increasing.
(b) Since the x−value 2 is greater than or equal
to 2, use the bottom part of the function to
find f (2), that is, f (x) = −3x + 6.
(2, ∞)
f (2) = −3(2) + 6 = 0
(b) In which interval, if any, is the function
decreasing?
(c) Since the x−value 3 is greater than or equal
to 2, use the bottom part of the function to
find f (2), that is, f (x) = −3x + 6.
From left to right, if the graph is going down,
it is decreasing.
f (3) = −3(3) + 6 = −3
(−∞, 0)
(d) Graph the function.
(c) In which interval, if any, is the function a
constant?
y
From left to right, if the graph stays at the
same level, it is constant.
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(0, 2)
(d) Domain: (−∞, ∞)
(e) Range: [1, ∞)
(f) Is the function graphed even, odd or neither? (even / odd / neither )
If the graph is symmetric with respect to the
y−axis, it is an even function.
If the graph is symmetric with respect to the
origin, it is an odd function.
4
5
3
2
1
.
-6
-5
-4
-3
-2
-1
-2
-3
-4
2. Determine whether the function is even, odd or
neither.
-5
-6
4
x
1
-1
2
3
4
5
6
MTH-112 Quiz 3 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1. For the function f (x) = −9x + 99, find:
5. Find the relative minimum and relative maximum of the graph, if they exist.
(a) To find f (x + h), replace x with x + h.
f (x + h) = −9(x + h) + 99
y
= −9x − 9h + 99
5
4
f (x + h) − f (x)
h
−9x − 9h + 99 − (−9x + 99)
=
h
−9x − 9h + 99 + 9x − 99
=
h
−9h
=
h
= −9
(−1, 3)
(b) The difference quotient:
(1, 3)
3
f (x)
2
1
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
-3
(−1, −3)
2. The graph of a function f (x) is given below.
Graph g(x) = f (x − 2) − 1 on the same coordinate plane.
-4
-5
The relative minimum is (write your answer as
an ordered pair):
y
6
5
A relative minimum is a point on the graph where
the graph turns from decreasing to increasing.
4
3
f (x)
2
(−3, −3)
g(x)
1
The relative maximum is (write your answer as
an ordered pair):
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
A relative maximum is a point on the graph where
the graph turns from increasing to decreasing.
-2
-3
-4
(1, 3)
3. Is the function f (x) graphed in problem 2 even,
odd or neither? ( even / odd / neither)
6. Is the function f (x) graphed in problem 5 even,
odd or neither? (even / odd neither )
The point (2, 4) is on the graph. The symmetric
point about the y−axis, which is (−2, 4), is also
on the graph. Therefore the graph is symmetric
with respect to the y−axis. Thus, the function is
even.
The point (1, 3) is on the graph. But the symmetric point about the y−axis, which is (−1, 3),
is not on the graph. Therefore the graph is not
symmetric with respect to the y−axis. Thus, the
function is not even.
4. (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.)
The point (1, 3) is on the graph. But the symmetric point about the origin, which is (−1, −3),
is not on the graph. Therefore the graph is not
symmetric with respect to the origin. Thus, the
function is not odd.
The symmetric point of (2, 3)with respect to
y−axis must also be on the graph.
(−2, 3)
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MTH-112 Quiz 4 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1. For the function f (x) = −4x2 − 3x + 7, find the
f (x + h) − f (x)
difference quotient:
h
(b) g(x) =
√
x + 10
Under the square root must be non-negative.
f (x + h)
x + 10 ≥ 0
= −4(x + h)2 − 3(x + h) + 7
x ≥ −10
= −4(x2 + 2hx + h2 ) − 3x − 3h + 7
Domain = [−10, ∞)
= −4x2 − 8hx − 4h2 − 3x − 3h + 7
(c) f (x) = x5 + 3x3 + 5
f (x + h) − f (x)
h
−4x2 − 8hx − 4h2 − 3x − 3h + 7 + 4x2 + 3x − 7
=
h
−8hx − 4h2 − 3h
=
h
h(−8x − 4h − 3)
=
h
= −8x − 4h − 3
Domain of any polynomial is all real numbers.
Domain = (−∞, ∞)
4. Let f (x) = 7x + 4 and g(x) = x2 − 6x. Find
the following. Simplify.
(a) (f + g)(x) is the sum of f (x) and g(x).
(f + g)(x) = f (x) + g(x)
2. The graph of a function f (x) is given below.
Graph g(x) = −f (x + 1) on the same coordinate plane.
= (7x + 4) + (x2 − 6x)
= x2 + x + 4
y
(b) (f −g)(3) is the difference between f (3) and
g(3).
5
4
3
f (x)
(f − g)(3) = f (3) − g(3)
2
= [7(3) + 4] − (3)2 − 6(3)
1
= 34
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
(c) (f g)(x) is the product of f (x) and g(x).
-2
(f g)(x) = [f (x)] [g(x)]
-3
-4
= (7x + 4)(x2 − 6x)
-5
= 7x3 − 38x2 − 24x
3. Find the domain of the following functions.
(Write your answers as intervals.)
f
(d)
(4) is the quotient of f (4) and g(4).
g
f
f (4)
(4) =
g
g(4)
7(4) + 4
=
(4)2 − 6(4)
= −4
1
−x + 2
The domain is all real numbers except 2.
(The number 2 makes the bottom zero.)
(a) f (x) =
Domain = (−∞, 2) ∪ (2, ∞)
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MTH-112 Quiz 4 - Solutions
(e) To find (f ◦ g)(x), substitute g function in
f function.
(f) To find (f ◦ g)(1), replace x with 1 in
(f ◦ g)(x) = 7x2 − 42x + 4.
(f ◦ g)(x) = f (g(x))
(f ◦ g)(1) = 7(1)2 − 42(1) + 4
= 7(x2 − 6x) + 4
= −31
= 7x2 − 42x + 4
MTH-112 Quiz 5
1. Find the center and the radius of the circle
graphed below.
5. If y varies directly as x, and y = 10 when x = 5,
find y when x = 39.
y
6
5
4
3
2
1
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
6. If y varies inversely as x, and y = 3 when
x = 33, find y when x = 15.
-2
Center:
Radius:
2. Find the center and the radius of the circle
(x + 1)2 + y 2 = 5.
Center:
Radius:
3. Write the equation of the circle, centered at
(10, 4), and of radius 9.
7. y varies directly as x and inversely as z 2 . If y is
64 when x = 16 and z = 6, find y when x = 2
and z = 5.
4. Find the center and the radius of the circle
x2 + y 2 − 4x + 2y − 11 = 0. Write the equation
in standard form.
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