Download MTH-112 Quiz 10

MTH-112 Quiz 10
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
1. Solve x2 − 4x + 3 > 0. Graph the solution set
on a real number line. Write the solution set in
interval notation.
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
6
3. Find the inverse of f (x) = 5x + 1.
5x − 3
≤ 4. Graph the solution set on
x−1
a real number line. Write the solution set in
interval notation.
x+3
inverse
4. Are f (x) = 5x − 3 and g(x) =
5
functions of each other? Show your work to
justify your answer.
2. Solve
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
6
1
MTH-112 Quiz 10 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
5x − 3
≤ 4. Graph the solution set on
x−1
a real number line. Write the solution set in
interval notation.
1. Solve x2 − 4x + 3 > 0. Graph the solution set
on a real number line. Write the solution set in
interval notation.
2. Solve
First, find the zeros of the equation
x2 − 4x + 3 = 0.
First, simplify the rational inequality to a rational
expression on one side and zero on the other side.
x2 − 4x + 3 = 0
(x − 3)(x − 1) = 0
5x − 3
x−1
5x − 3 4
−
x−1
1
5x − 3 4(x − 1)
−
x−1
1(x − 1)
5x − 3 − 4x + 4
x−1
x+1
x−1
x−3=0
x−1=0
x=3
x=1
Separate the number line into three intervals using
the two points x = 1 and x = 3.
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
≤4
≤0
≤0
≤0
≤0
Now find the zeros of the top polynomial and the
bottom polynomial.
6
x−1=0
x+1=0
Pick three test points, one from each interval, and
check whether the factored form of the polynomial
is a positive or negative value. We are interested
in positive values, because we are looking for ≥ 0.
x = −1
Separate the number line into three intervals using
the two points x = −1 and x = 1.
?
Test Pts
(x − 3)(x − 1)
≥0
0
(0 − 3)(0 − 1) = (−)(−)
= (+)X
2
(2 − 3)(2 − 1) = (−)(+)
= (−)×
4
(4 − 3)(4 − 1) = (+)(+)
= (+)X
-6 -5 -4 -3 -2 -1 0
×
-6 -5 -4 -3 -2 -1 0
1
2
X
3
4
5
1
2
3
4
5
6
Pick three test points, one from each interval, and
check whether the rational expression is a positive
or negative value. We are interested in negative
values, because we are looking for ≤ 0.
The test points 0 and 4 yield positive values.
X
x=1
Test Pts
x+1
x−1
≤0
−2
−2 + 1
(−)
=
−2 − 1
(−)
= (+)×
0
(+)
0+1
=
0−1
(−)
= (−)X
2
2+1
(+)
=
2−1
(+)
= (+)×
?
6
The interval notation: (−∞, 1) ∪ (3, ∞)
1
MTH-112 Quiz 10 - Solutions
The test point 0 yields a negative value.
×
y = 5x − 3
×
X
-6 -5 -4 -3 -2 -1 0
First, replace f (x) with y.
1
2
3
4
5
6
Swap x and y.
x = 5y − 3
The interval notation: [−1, 1). (1 is not in the
solution because 1 is a zero of the denominator of
the rational expression.)
Solve for y.
x = 5y − 3
3. Find the inverse of f (x) = 5x + 1.
y = 5x + 1
x + 3 = 5y
x+3
=y
5
Swap x and y.
Replace y with f −1 (x).
First, replace f (x) with y.
x = 5y + 1
f −1 (x) =
Solve for y.
Since f −1 (x) = g(x), f and g functions are inverses of each other.
x = 5y + 1
x − 1 = 5y
x−1
=y
5
Yes.
Second method: Find (f ◦ g)(x), and see whether
it is equal to x.
Replace y with f −1 (x).
f −1 (x) =
x+3
5
x−1
5
(f ◦ g)(x) = f (g(x))
x+3
=5
−3
5
x+3
4. Are f (x) = 5x − 3 and g(x) =
inverse
5
functions of each other? Show your work to
justify your answer.
=x+3−3
=x
There are two methods to do the problem. You
may use either method.
Since (f ◦ g)(x) = x, f and g functions are inverses of each other.
First method: Find the inverse of f function, and
see whether it is equal to g function.
Yes.
2
MTH-112 Quiz 11
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
5. Approximate the number e−0.77 using a calculator (round to three decimal places).
1. Solve x(x2 + 1) ≥ 0. Graph the solution set on
a real number line. Write the solution set in
interval notation.
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
6
6. The graph of f (x) = 2x is given below. Graph
g(x) = 2x − 2, and answer (a) - (d).
y
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
2012
≥ 0. Graph the solution set on
2. Solve 2
x + 16
a real number line. Write the solution set in
interval notation.
1
2
3
4
5
6
x
-2
-3
-4
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
-5
6
-6
(a) The domain of g(x):
(b) The range of g(x):
(c) The equation of the horizontal asymptote
of g(x):
3. Find the inverse of f (x) =
5
x + 3.
2
(d) The y−intercept of g(x):
7. An initial investment of $ 1234.56 is appreciated
for 8 years in an account that earns 7% interest,
compounded continuously. Find the amount of
money in the account at the end of the period.
(Round answer to the nearest cent.)
4. Approximate the number 20.77 using a calculator (round to three decimal places).
1
MTH-112 Quiz 11 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1. Solve x(x2 + 1) ≥ 0. Graph the solution set on
a real number line. Write the solution set in
interval notation.
The top and bottom of the rational inequality
have no zeros. Therefore the value of the rational expression x2012
2 +16 is either always positive or
always negative. Since 2012 is a positive number
and x2 + 16 is positive for any x value, x2012
2 +16 is
always positive. Therefore, for any x value, the
inequality is always true.
First, find the zeros of the equation
x(x2 + 1) = 0.
x(x2 + 1) = 0
x=0
x2 + 1 = 0
x2 = −1
x=0
-6 -5 -4 -3 -2 -1 0
Since x2 is always nonnegative it cannot be equal
to −1. So the only solution of the equation is 0.
Separate the number line into two intervals using
the point x = 0.
1
2
3
4
5
6
The interval notation: (−∞, ∞)
3. Find the inverse of f (x) =
5
x + 3.
2
First, replace f (x) with y.
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
6
y=
5
x+3
2
Swap x and y.
Pick two test points, one from each interval, and
check whether the factored form of the polynomial
is a positive or negative value. We are interested
in positive values, because we are looking for ≥ 0.
x=
5
y+3
2
Solve for y.
Test
?
x−3=
Pts
x(x2 + 1)
≥0
−1
(−1)[(−1)2 + 1] =
(−)(+) = (−)×
1
(1)[(1)2 + 1] =
(+)(+) = (+)X
5
y
2
2
(x − 3) = y
5
2x − 6
=y
5
Replace y with f −1 (x).
The test point 1 yields a positive value.
f −1 (x) =
×
-6 -5 -4 -3 -2 -1 0
2x − 6
5
X
1
2
3
4
5
4. Approximate the number 20.77 using a calculator (round to three decimal places).
6
1.705
The interval notation: [0, ∞)
5. Approximate the number e−0.77 using a calculator (round to three decimal places).
2012
2. Solve 2
≥ 0. Graph the solution set on
x + 16
a real number line. Write the solution set in
interval notation.
0.463
1
MTH-112 Quiz 11 - Solutions
6. The graph of f (x) = 2x is given below. Graph
g(x) = 2x − 2, and answer (a) - (d).
(−2, ∞)
(c) The equation of the horizontal asymptote
of g(x):
y
6
The equation of the horizontal asymptote of
an exponential function is always y = the
number used when writing the range.
y = −2
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
(d) The y−intercept of g(x):
x
g(0) = 20 − 2 = −1
-2
(0, −1)
-3
-4
-5
7. An initial investment of $ 1234.56 is appreciated
for 8 years in an account that earns 7% interest,
compounded continuously. Find the amount of
money in the account at the end of the period.
(Round answer to the nearest cent.)
-6
(a) The domain of g(x):
The domain of any exponential function is
always all real numbers.
(−∞, ∞)
A = P ert
(b) The range of g(x):
= 1234.56e0.07·8
Because 2 is subtracted, the range is from
−2 to positive infinity (−2 is not included).
= $2161.31
2