Answer:f(x)=x8−8x6+19x4−12x3+14x2−8x+9=x8−8x6+16x4+3x4−12x3+12x2+2x2−8x+8+1
which allows us to rewrite f(x) as
f(x)=x4(x2−4)2+3x2(x−2)2+2(x−2)2+1
The first three terms are clearly non-negative, and each reaches their minimum of 0 at x=2 (the first term also has a minimum at x=−2). Thus, the minimum of f(x) must be 1.
This can't really be generalized. (I mean, you can apply the approach generally, but it won't generally give you such a convenient result.) I'm not sure I would have looked for this decomposition of f(x) except for the presence of the question.
Step-by-step explanation:
Denasia and Kenya are training to run a half marathon. In the first week of training, Denasia runs 11 miles and Kenya runs 14 miles.
Each week, Denasia increases her weekly total by 2 miles, and Kenya increases her weekly total by 1.5 miles. Write and solve an
equation to find how many weeks after the first week of training Denasia and Kenya will be running the same number of miles.
A. 2w + 14 = 1.5w+11; w=-6 weeks
B. 2w + 11 = 1.5w+14; w=6 weeks
C. 11w+2= 14w +1.5; w=0, 167 weeks
Option B. 2w + 11 = 1.5w + 14, w = 6 weeks is the correct answer
Miles run by Denasia = 11 miles
Miles run by Kenya = 14 miles
Let w represent the number of weeks after which both Kenya and Denasia will be running the same number of miles
Formulating the equation we get:
Miles run by Denasia + Increase in miles each week*Number of weeks = Miles run by Kenya + Increase in miles each week*Number of weeks
11 + 2w = 14 + 1.5w
0.5w = 3
w = 6
Hence, option B is the correct answer according to the given contraints and both Denasia and Kenya will run the same distance in the 6th week
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Can someone help me with this please?
Step-by-step explanation:
in general, a linear function is always of the structure
y = ax + b
"a" is the slope of the line and is the ratio (y coordinate difference / x coordinate difference) when going from one point on the line to another.
"b" is the y-intercept (the y-value when x = 0).
so, if we don't see the solution right away with experience, we have to use the data points (usually starting with the first 2) to solve them 2 equations with 2 variables a, b :
2 = a×1 + b = a + b
1 = a×2 + b = 2a + b
now we can simply subtract the first equation from the second and get
1 = 2a + b
- 2 = a + b
------------------
-1 = a + 0
a = -1
1 = 2×-1 + b = -2 + b
b = 3
and the equation is
y = -x + 3
therefore, when the input is n (that means f(n))
y = output = -n + 3
Compare these two fractions: 3/10 and 4/5*03/10 > 4154/5>3/10О4/5 = 3/10
4x + 10 =-26 solve for x
Answer:
x=-9Step-by-step explanation:
To solve for x, isolate this equation from left to right.
4x+10=-26
First, subtract by 10 from both sides.
4x+10-10=-26-10
Solve.
-26-10=-36
4x=-36
Then, you divide by 4 from both sides.
4x/4=-36/4
Solve.
-36/4=-9
[tex]\Rightarrow \boxed{\sf{x=-9}}[/tex]
As a result, the solution is x=-9, which is the correct answer.
I hope this helps, let me know if you have any questions.
Answer:
-9
Step-by-step explanation:
First you subtract 10 from each side
4x + 10 = -26
-10 -10
then you divide by your x
4x = -36
4 4
the 4s cancel out leaving you with x only
x = -36/4
x = -9
-7 -6 -5 -4 -3 -2 ONO MN ++ HA -6-5-4-3-2-1 + 11 2 3 4 5 6 7 8 9 -2 -3 F-4 -5 -6 -7 -A + jo voo Alcon ixth If this is the graph of f(-x) = a +k, then : A. 0 < a < 1 B. a < 0 O c. a> 1 O D. K> 1
The function is
[tex]f(x)=a^{(x+h)}+k[/tex]The limit when x->+/- infinite are (analitically)
[tex]\begin{gathered} \lim _{x\to\infty}f(x)=a^{(\infty+h)}+k=a^{\infty}+k \\ \text{and} \\ \lim _{x\to-\infty}f(x)=a^{(-\infty+h)}+k=\frac{1}{a^{\infty}}+k \\ \end{gathered}[/tex]And, from the figure,
[tex]\begin{gathered} \lim _{x\to\infty}f(x)=\infty \\ \text{and} \\ \lim _{x\to-\infty}f(x)=-4 \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} \Rightarrow a^{\infty}+k=\infty \\ \Rightarrow a>1 \\ \text{and} \\ \Rightarrow-4=\frac{1}{a^{\infty}}+k,a>1 \\ \Rightarrow-4=k \end{gathered}[/tex]Therefore, the answer is option C, a>1.
Which functions are equivalent to f (x) = RootIndex 4 StartRoot 162 EndRoot Superscript x? Check all that apply.
The functions that are equivalent are
f (x) = 162 Superscript StartFraction x Over 4f (x) = (3 RootIndex 4 StartRoot 2 EndRoot) Superscript xf (x) = left-bracket 3 (2 Superscript one-fourth Baseline) right-bracket Superscript xWhat are roots of a number?
The root of a number is the inverse of the exponents. for instance
a squared has an inverse of square root of a.
mathematically:
a^2 = (a)^1/2 has an inverse of √a
How to find the equivalents of the given dataThe data given:
[tex]\sqrt[4]{162^{x} }[/tex]
Solving the given data for the equivalence
[tex]\sqrt[4]{162^{x} }=162^{x/4}[/tex]
The above proves option A correct
[tex]\sqrt[4]{(3*3*3*3*2)^{x} }[/tex]
[tex]\sqrt[4]{(3^{4} *2)^{x} }[/tex]
[tex](3\sqrt[4]{(2) })^{x}[/tex]
The above solution makes option B correct
[tex](3\sqrt[4]{(2) })^{x}=(3{(2)^{1/4} })^{x}[/tex]
The above proves Option D correct
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complete question
Which functions are equivalent to f (x) = RootIndex 4 StartRoot 162 EndRoot Superscript x? Check all that apply.
f (x) = 162 Superscript StartFraction x Over 4
f (x) = (3 RootIndex 4 StartRoot 2 EndRoot) Superscript x
f (x) = 9 RootIndex 4 StartRoot 2 EndRoot Superscript x
f (x) = 126 Superscript StartFraction 4 Over x
f (x) = left-bracket 3 (2 Superscript one-fourth Baseline) right-bracket Superscript x
PLEASE HELP ASAP!!!!!!!
Rotate R and reflect 90
Step-by-step explanation:
Learn with an example
or Watch a video D
Two cars leave the same parking lot, with one heading north and the other heading east.
After several minutes, the northbound car has traveled 3 kilometers, and the eastbound car
has traveled 4 kilometers. Measured in a straight line, how far apart are the two cars?
kilometers.
The two cars are 5 kilometers far apart from each other.
What is Pythagoras theorem?
The Pythagoras theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.
Given that;
Two cars leave the same parking lot, with one heading north and the other heading east.
After several minutes, the northbound car has traveled 3 kilometers, and the eastbound car has traveled 4 kilometers.
Now,
The straight path is find by using the Pythagoras theorem as;
Straight path = √3² + 4²
= √9 + 16
= √25
= 5 km
Therefore,
The two cars are 5 kilometers far apart from each other.
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PLEASE HELP
Classify the equation 6x + 4x-1= 2(5x + 4) as having one solution, infinitely many solutions, or no solution by
solving the problem. Solve the equation for x then classify the solution.
Answer: NO SOLUTION
Step-by-step explanation:
6x + 4x - 1 = 2(5x + 4)
10x - 1 = 2(5x+4)
10x - 1 = 10x + 8
-1 = 8
NO SOLUTION
When going out for dinner, Sally left an 18% tip. The tip was 9.50. How much was the dinner bill before the tip?
The dinner bill of Sally before the tip is $53.
Given that, Sally left an 18% tip. The tip was 9.50.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Let the dinner bill of Sally be x.
Now, 18% of x = 9.50
⇒ 18/100 × x = 9.50
⇒ 0.18x = 9.50
⇒ x = 9.50/0.18
⇒ x = 52.77
≈ 53
Therefore, the dinner bill of Sally before the tip is $53.
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At the beginning of spring, Anand planted a small sunflower in his
backyard. The sunflower's height in inches, h, after w weeks, is given by the equation h = 3.75w+18. What
could the number 3.75 represent in the equation?
Hint: this equation is in slope-intercept form: y=mx+b
The rate of change in the sunflower's height
The sunflower's height after one week
The sunflower's height when it is planted
Answer:
3.75 represents how much the flower grows every week.
Step-by-step explanation:
Can someone please help me! I have no idea what I am doing and have no book to look from!
Answer:
Step-by-step explanation:
1) Find DGE
Bisecting an angle makes it half. A straight line is 180
180/2 = DGE
90 = DGE
2) Find FGE
Bisecting an angle makes it half. A straight line is 180
180/2 = FGE
90 = FGE
42
X
40
Find the unknown side length, x. Write your answer in simplest radical form.
A. 241
B. 4 29
C. 48
D. 58
The unknown side length of the right triangle is 58 units.
What is pythagoras theorem for a right triangle?
In essence, the Pythagorean theorem is used to determine a triangle's angle and length of an unknown side. In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse. Due to its position opposite the 90° angle, the hypotenuse in this case is the longest side. When the positive integer sides of a right triangle (let's say sides a, b, and c) are squared, the result is an equation known as a Pythagorean triple.
Mathematically, Hypotenuse² = Base² + Perpendicular²
Given, the length of the perpendicular of the right triangle = 42 units
Also, the length of the base of the right triangle = 40 units
Let the length of the hypotenuse of the right triangle = 'x' units
Following the statement of the pythagoras theorem as established in literature, we have: Hypotenuse² = Base² + Perpendicular²
Substituting the values from the question given, we get:
x² = 42² + 40² = 3364 ⇒ x = √3364 ⇒ x = √58² ⇒ x = 58
Therefore, the unknown side length of the right triangle is 58 units.
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i'm still confused. helpppppppppppppppppppppppppppppppppppppppppppppppppppppp meeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Answer:
I believe you forgot to add the attachment
Step-by-step explanation:
There is no math problem I can see to help you
3. a person is watching the space shuttle launch. the person is 3000 ft from the launch pad. how fast is the distance between the person and the shuttle changing when the shuttle is 4,000 ft high and rising at a rate of 800 ft/sec?
The distance between the person and the shuttle is changing at a rate of 640 feet/sec when the shuttle is 4000 ft high and is rising at a rate of 800 ft/sec.
Considering x to be the height of the triangle and y to be the hypotenuse of the triangle, a right-angled triangle will be formed according to the given information having a base of 3000 feet which is the distance of the person from the launch pad.
In this right-angle triangle x will be the height of the shuttle and y will be the distance between the person and the shuttle.
Now we apply the Pythagorean theorem to the triangle;
y² = x²+(3000)²
Now differentiating this equation with respect to time,t :
2y (dy/dt) = 2x(dx/dt) + 0
dy/dt = x(dx/dt)/y
As the shuttle is rising at a speed of 800 ft/sec, (dx/dt)=800
Substitute 4000 for x into the equation, y²=x²+(3000)², to find y;
y² = (4000)²+(3000)²
y = 5000
Now we substitute 4000 for x, 5000 for y, and 800 for dx/dt in the equation, dy/dt = x(dx/dt)/y
dy/dt = 4000(800) / 5000
dy/dt = 640
Therefore, the distance between the person and the shuttle is increasing at a rate of 640 feet/sec.
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2. Find the slope and y-intercept of the line.
O slope=3 y-intercept=2
O slope 2 y-intercept=3
O slope 3/2 y-intercept=2
O slope 2/3 y-intercept=2
Identify the factors of x2 − 5x − 24. (1 point) (x + 8)(x − 3) (x − 8)(x + 3) (x + 4)(x − 6) (x − 4)(x + 6)
Answer:
(x+3)(x–8)
Step-by-step explanation:
Normally you will always write factors in this form: (variable + smallest number)(variable – largest number)
Correct form: (x+2)(x–8)
Incorrect form: (x–2)(x+8)
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Find the slope of the line that passes through (1, 6) and (3, 7)
[tex]m = \frac{y2 - y1}{x2 - x1} \\ m = \frac{7 - 6}{3 - 1} \\ m = \frac{1}{2} [/tex]
I ALSO PROVIDED YOU WITH THE FORMULA OF FINDING THE GRADIENT/SLOPE FOR FUTURE USE
HOPE THIS HELPS.
Answer: Y= 1/2x + 5.5
14/3 = 7y solve for y and simplify your answer as much as possible
Answer:
2/3 = y
Step-by-step explanation:
to solve you got to cancel out the 7
so 7y/7.
whatever you do to one side, you do to the other.
14/3 ÷ 7 = 7y/7
14/3 ÷ 7 = 0.6666... = 2/3 = y
Is this relation a function? Justify your answer. 10 9 8 7 5 3 2 1 1 2 3 4 5 6 7 8 9 10 11 A. Yes, because the number of x-values is the same as the number c y-values. B. No, because two points with the same x-value have different y- values. C. Yes, because every x- and y-value is positive. f se two points with the same vevalue have different x-
The correct answer is option B that is No, because two points with the same x-value have different y values.
A relation in mathematics may be defined as the term which describes the relationship between the input and output variables. A function may be defined as the expression in which for one value of input variable x there is only one output variable y. The input variable is called independent variable and output variable is called dependent variable. From the graph given in question it can be seen that at point x = 2 there are two points on the y axis that are y = 2 and y = 11. So, this violates the basic definition of function as for input x there are two outputs y and hence it cannot be regarded as function.
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work out the value of 6 2 plus 3 3
Answer:95
Step-by-step explanation:
This can simply be answered by adding the first digit of both numbers which would be 9 then add the secound number of both numbers and that would be 5
The answer is 95
Answer:
if its 6^2 + 3^3 then its 63
Step-by-step explanation:
6^2 simplified is 36 and 3^3
6 x 6 = 36 + 27 = 63
3 x 3 x 3 =27
3 x 3= 9
9 x 3= 27
The next two problems show the attempts of two different students to provide a proof of the same statement. Each proof has something about it that is not quite correct. Explain how you would improve the argument.
We will improve the argument by stating that two intersecting lines have vertically opposite angles equal in magnitude.
We are given a diagram in which we have two lines, namely "m" and "n".The intersection point of the two lines is P.We need to prove that ∠1≅∠3.We know that line "m" is a straight line, so it makes a total angle of 180° on either of its sides.∠1+∠2 = 180°We know that line "n" is a straight line, so it makes a total angle of 180° on either of its sides.∠3+∠2 = 180°From the above two equations, we find that :∠1+∠2 = 180° = ∠3+∠2∠1+∠2 = ∠3+∠2∠1 = ∠3Thus, ∠1≅∠3.To learn more about lines, visit :
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What percentage of the number 1 to 20 contain the digit 2????
The amount as a percentage is 15% which represents the number 1 to 20 containing the digit 2.
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
We know that the numbers 1 to 20 that contain the digit 2 are 2,12, and 20.
So, the quantity of the numbers is 3
It’s out of 20 numbers, this means 3/20
Calculating percentages indicated above can be readily computed using the formula below:
Percentage = (Value/Total Value) × 100
The amount as a percent = (Value/Total Value) × 100
The amount as a percent = (3/20) × 100
The amount as a percent = 15%
Thus, the amount as a percentage is 15% which represents the number 1 to 20 containing the digit 2.
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0.194805194805...
Convert the decimal into a fraction.
Answer:
The fraction is 15/77Step-by-step explanation:
The repeated part is 194805, six digits. Let the fraction be x.
Convert as follows:
1000000x - x = 194805.194805 ... - 0.194805 ... 999999x = 194805x = 194805/999999Find prime factors of both numerator and denominator and simplify by cancelling common factors:
194805 = 3⁴*5*13*37,999999 = 3³*7*11*13*37.The common factors are:
3³*13*37When they cancel we are left with:
x = (3*5)/(7*11) x = 15/77identify the paragraph proof for the two-column proof. Given: m∠d=125° and m∠r=125° Prove: ∠d≅∠r Two-Column Proof 1. m∠d=125° (Given) 2. m∠r=125° (Given) 3. m∠d=m∠r (Trans. Prop. of = ) 4. ∠d≅∠r (Def. of ≅∠s)
The two column proof is written as follows
Statement Reason
1. m ∠ d = 125° Given
2. m ∠ r = 125° Given
3. m ∠ d = m ∠ r Definition of equality
4. ∠ d ≅ ∠ r Definition of congruent angles
What is two column proof?This is a method of proof used in geometry. It helps to show how the required proof came to be possible. The major items in two column proof is:
StatementReasonwhat is equality?The term equality as used in mathematics is defined to mean that the terms in comparison can be replace by the other
Using the given question as an instance:
m ∠ d = 125° and m ∠ r = 125°
This means that the two angles have same value and they can replace each other.
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8.
S
-80
ZX D
lo
Please help
HELP PLEASE
Answer:
[tex]\begin{aligned}&\phantom{=.} \dfrac{1}{8}x \cdot \dfrac{-4}{5}x\\\\&=\dfrac{1}{2} \cdot \dfrac{\boxed{-1}}{5} \cdot x \cdot \boxed{x}\\\\& = \dfrac{\boxed{-1}}{\boxed{10}} \;x\;^\boxed{2}\end{aligned}[/tex]
Step-by-step explanation:
According to the Commutative Property Law of Multiplication, changing the order or position of the numbers does not change the end result.
Therefore, collect like terms by moving the fractions to the left and the variables to the right.
As the denominator of the first fraction has been divided by 4, the numerator of the second fraction should also be divided by 4.
Therefore:
[tex]=\dfrac{1}{2} \cdot \dfrac{\boxed{-1}}{5} \cdot x \cdot \boxed{x}\\[/tex]
[tex]\textsf{Apply\;the\;fraction\;rule} \quad \dfrac{a}{c}\cdot \dfrac{b}{d}=\dfrac{ab}{cd}.[/tex]
[tex]\textsf{Apply\;the\;exponent\;rule}\quad \:aa=a^2.[/tex]
Therefore:
[tex]=\dfrac{\boxed{-1}}{\boxed{10}} \;x\;\boxed{^2}[/tex]
Which number is the smallest?
pls answer fast
Responses
A 4.68 x 10−44.68 x , 10 -4
B 5.48 x 10−85.48 x , 10 -8
C 2.85 x 10−62.85 x , 10 -6,
D 3.24 x 10−53.24 x , 10 -5,
E 1.28 x 10−4
The smallest number from the given options is
[tex]5.48 × {10}^{ - 8} [/tex]
The correct answer option is option d
How to find the smallest number?A.
[tex]1.28 × {10}^{ - 4} [/tex]
= 1.28 × 0.0001
= 0.000128
B.
[tex]2.85 × {10}^{ - 6} [/tex]
= 2.85 × 0.000001
= 0.00000285
C.
[tex]3.24 × {10}^{ - 5} [/tex]
= 3.24 × 0.00001
= 0.0000324
D.
[tex]5.48 × {10}^{ - 8} [/tex]
= 5 48 × 0.00000001
= 0.0000000548
Therefore, the smallest number is
[tex]5.48 × {10}^{ - 8} [/tex]
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Which exponential equation is equivalent to this logarithmic equation? \log _(5)x - \log _(5)25=7
The exponential equation [tex]14=5^x[/tex] is equivalent to the given logarithmic equation.
Which is the Exponential function?The exponential function that is represented as y=[tex]m^x[/tex], where:
m=base, m>0
x= exponent
Logarithm PropertiesKnowing some of the main logarithm rules.
Product Rule with the same base: you should repeat the base add the logarithms of the factors.example: [tex]log(a*b)= log a + log b[/tex]
Quotient Rule: you should subtract the logarithm of the numerator with the logarithm of the denominator.example: [tex]log\frac{a}{b} = log a - log b[/tex]
Power Rule: you should multiply the exponent by the logarithm of the base
example: [tex]log\frac{a^b} = b*log a[/tex]
For solving this question, you should apply the logarithm rules and rewrite the function as an exponential equation.
[tex]log_5(x)}-{log_5(25)=7[/tex]
[tex]\frac{log_5(x)}{log_5(25)} =7[/tex]
[tex]{log_5(x)}=7{log_5(25)}[/tex]
[tex]{log_5(x)}=7{log_5(5^2)}[/tex]
[tex]{log_5(x)}=7*2[/tex]
[tex]{log_5(x)}=14[/tex]
[tex]x=5^{14^}[/tex]
Therefore, the exponential function is [tex]14=5^x[/tex]
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Tanya sells cars. Her yearly salary is $35,000 plus 8% of her sales. Type and solve an inequality to determine her necessary sales to earn over $50,000.
Tanya yearly salary is
$35000 + 8% of her sales
Let her sales be x
Since, her sales is x
35000 + 0.08x > salary
To determine her necessary sales to earn over $50, 000
Let $50, 000 be her benchmark salary
35,000 + 0.08 * x > 50000
35000 + 0.08x > 50000
0.08x > 50000 - 350000
0.08x > 15000
Divide both sides by 0.08
0.08x/0.08 > 15000/0.08
x > 15000 / 0.08
x > 187,500
Therefore, she will need a sales of 188,000 and above to earn above $50,000