The value of the composite function g(f(x)) is 1 / 3 (81x² - 216x + 144) + 4
How to solve composite function?A composite function is a function that depends on another function. In a composite function, the output of one function becomes the input.
Therefore, let's solve the function as follows;
f(x) = 1 / 3 x² + 4
g(x) = 9x - 12
The value of g(f(x)) can be found as follows:
To find g(f(x)) we have to substitute the f(x) in g(x).
Therefore,
g(f(x)) = 1 / 3 (9x - 12)² + 4
(9x - 12)(9x - 12) = 81x² - 108x - 108x + 144
(9x - 12)(9x - 12) = 81x² - 216x + 144
Therefore,
g(f(x)) = 1 / 3 (81x² - 216x + 144) + 4
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Find the zeros by using the quadratic formula and tell whether the solutions are real or imaginary. F(x)=x^2–8x+2
We have to calculate the zeros of the function with the quadratic formula.
[tex]f(x)=x^2-8x+2[/tex][tex]\begin{gathered} x=\frac{-(-8)}{2\cdot1}\pm\frac{\sqrt[]{(-8)^2-4\cdot1\cdot2}}{2\cdot1}=\frac{8}{2}\pm\frac{\sqrt[]{64-8}}{2}=4\pm\frac{\sqrt[]{56}}{2}=4\pm\sqrt[]{\frac{56}{4}}=4\pm\sqrt[]{14} \\ \\ x_1=4+\sqrt[]{14}\approx4+3.742=7.742 \\ x_2=4-\sqrt[]{14}\approx4-3.742=0.258 \end{gathered}[/tex]The roots are x1=7.742 and x2=0.258, both reals., both
Four more than three times a number, is less than 30. Which of the following is not a solution?61278
Solution
- To solve the question, we simply need to interpret the question line by line.
- Let the number be x.
- "Four more than three times a number" can be written as:
[tex]\begin{gathered} \text{ Three times a number is: }3x \\ \text{ For more than three times a number becomes: }4+3x \end{gathered}[/tex]- "Four more than three times a number is less than 30" can be written as:
[tex]4+3x<30[/tex]- Now, we can proceed to solve the inequality and find the appropriate range of x. This is done below:
[tex]\begin{gathered} 4+3x<30 \\ \text{ Subtract 4 from both sides} \\ 3x<30-4 \\ 3x<26 \\ \text{ Divide both sides by 3} \\ \frac{3x}{3}<\frac{26}{3} \\ \\ \therefore x<8\frac{2}{3} \end{gathered}[/tex]- This means that all correct solutions to the inequality lie below 8.666...
- This further implies that any number greater than this is not part of the solutions of the inequality.
- 12 is greater than 8.666
Final Answer
The answer is 12
What is the greatest common factor of 9 and 72?
The Greatest Common Factor of 9 and 72 is: 9
SOLUTION
Problem Statement
The question asks us to find the greatest common factor of 9 and 72.
Method
In order to solve this question, we just need to follow these steps:
1. Write out the prime factors of 9 and 72
2. Choose the common factors from both expressions.
3. Multiply the common factors.
Implementation
1. Write out the prime factors of 9 and 72:
[tex]\begin{gathered} 9=1\times3\times3 \\ \text{The common factors of 9 are: 3 and 3} \\ \\ 72=1\times2\times2\times2\times3\times3 \\ \text{Common factors of 72 are: 1,2, 2, 2 and 3, 3} \end{gathered}[/tex]2. Choose the common factors from both expressions.:
We need to examine the two expressions for 9 and 72 above. Choose the common values.
[tex]\begin{gathered} 3\times3\text{ is common to both 9 and 72} \\ i\mathrm{}e\text{.} \\ 9\text{ is common to both 9 and 72} \\ 3\text{ is common to both 9 and 72 as well} \\ 1\text{ is also common to both 9 and 72} \end{gathered}[/tex]3. Multiply the common factors.:
[tex]\begin{gathered} \text{Thus, choosing the greatest values from 1,3 and 9.} \\ \therefore\text{The Greatest Common Factor = 9} \end{gathered}[/tex]Final Answer:
The Greatest Common Factor of 9 and 72 is: 9
3 A free diver can dive at a rate of -0.75 meters per second. About how long would it take to reach a depth of -145 meters?
Answer:193.33s or 3.2 mins
Step-by-step explanation:
Here the diver dives at -0.75 constant velocity. So we can use the formula s = vt
from there we can find out the time when s = -145 m and v = -0.75m/s
Four friends had dinner at a restaurant. The bill was $45.40, and they added a 20% tip. The 4 friends shared the total cost equally. How much money did each person pay?
Answer:
$13.62
Step-by-step explanation:
To find the tip of 20%, multiply 45.40 by 0.20, and you get 54.48.
Now, since there are 4 friends, divide the amount by 4.
54.48/4=13.62
So each person would pay $13.62.
Hope this helps! :D
If f (x) = 4x^3 - 25x^2 – 154x+ 40 and (x - 10) is a factor, what are the remaining factors?
It goes from -1 to 1 on the x axis.
ANSWER :
EXPLANATION :
Real numbersA stock lost 8 and 3/8 on Monday, 1 and 5/8 points on Tuesday. On Wednesday, it gained 13 points. What was the net gain or loss of the stocks for these three days?
Answer: The net gain was 3 stocks for these three days.
Explanation
Given
• Monday: lost 8 and 3/8.
,• Tuesday: lost 1 and 5/8 points.
,• Wednesday: gained 13 points.
Procedure
To calculate the net gain, we have to add all the values we are giving, considering that the stock lost will have a negative sign:
[tex]\text{Net gain}=Gains-losses[/tex]0. Calculating the losses
We have to mixed numbers, for which we can add the whole numbers and the fractions separately.
• Whole numbers
[tex]-8-1=-9[/tex]• Fractions
[tex]-\frac{3}{8}-\frac{5}{8}=-\frac{8}{8}=-1[/tex]At last, we add both results and we get the losses:
[tex]losses=-9-1=-10[/tex]2. Calculating the gains
As we only have one quantity, there is no need for calculations:
[tex]Gains=13[/tex]3. Calculating net gain
Finally, we put both quantities in the formula:
[tex]\text{Net gain}=13-10=3[/tex]in the graph below line k,y = -x makes a 45 degree angle with the X and Y axes complete the following
The point with a coordinate of (2,5) will be translated into y=-x line.
The transformation for y=-x would be:
1. x'= -y
2. y'= -x
For x=2 and y=5 would be:
x'= -y
x'= -5
y'= -x
y'= -2
The translated coordinate would be: (-5, -2)
X) *11.4.14 Find the volume of the cylinder in terms of it and to the nearest tenth. 2 in 1 in The volume in terms of it is V= in3
We can calculate the volume of the cylinder as the product of the area of the base and the height of the cylinder.
[tex]V=A_b\cdot h[/tex]The area of the base is equal to:
[tex]A_b=\pi r^2=\pi\cdot2^2=4\pi[/tex]Then, the volume becomes:
[tex]V=A_b\cdot h=4\pi\cdot1=4\pi\approx12.6\text{ in}^3[/tex]Answer:
The volume in function of π is V = 4π in^3.
The volume rounded to the nearest tenth is 12.6 in^3.
I’ve already done this problem, but I’m being told it’s wrong and I need to simplify but I don’t know how to do it with this question.
Rick buys a $4,500 stove with an installment plan that requires 12% down. How much
is the down payment?
O $540
O $375
O $250
O $125
Answer: 540
Step-by-step explanation: 12% of 4500 is 540
Andy spent 1/3 of his money on pastries and 3/4 of his remaining money on 2 pies. Each pie costs 6 times as much as each pastry. if all pastries cost the same how many did he buy
The area (in square inches) of a rectangle is given by the polynomial function A(b)=b^2 +9b+18. If the width of the rectangle is (b+3) inches what is the length?
As given by the question
There are given that area of rectangle and width of a rectangle
[tex]\begin{gathered} \text{Area}=A(b)=b^2+9b+18 \\ \text{Width}=(b+3) \end{gathered}[/tex]Now,
From the formula of area of a rectangle:
[tex]\text{Area}=\text{length}\times width[/tex]Then,
Put the value of an area and width into the above formula
So,
[tex]\begin{gathered} \text{Area}=\text{length}\times width \\ b^2+9b+18=length\times(b+3) \end{gathered}[/tex]Then,
[tex]\begin{gathered} b^2+9b+18=length\times(b+3) \\ (b+3)(b+6)=\text{length}\times(b+3) \\ \text{length}=\frac{(b+3)(b+6)}{(b+3)} \\ \text{length}=(b+6) \end{gathered}[/tex]Hence, the value of length is ( b + 6 ).
Sean johnson 1. The angles of a triangle are described as follows: angle A is the largest angle: its measure is twice the measure of angle B. The measure of angle C is 2 less than half the measure of angle B. Find the measures of the three angles in degrees.
Explanation
Let
angle A is the largest angle
[tex]m\measuredangle A=m\measuredangle A[/tex]its measure is twice the measure of angle B,( in other words you have to multiply angle b by 2, to get angle A)
[tex]2(m\measuredangle B)=m\measuredangle A[/tex]theres 2 fill in the blank boxes and 3 drop down menus, below i will list the options in the drop down menus.box 1 - apply quotient identities, apply Pythagorean identities, apply double-number identities, apply even-odd identities.box 2 - apply cofunction identities, use the definition of subtraction, apply even-odd identities, Write as one expresssion combine like terms.box 3 - apply cofunction identities, apply double-number identities, apply Pythagorean identities, apply even-odd identities.
Solution
Box 1 : Apply Quotient Identities
[tex]cotx-tanx=\frac{cosx}{sinx}-\frac{sinx}{cosx}[/tex]The answer for the first box is
[tex]\begin{equation*} \frac{cosx}{sinx}-\frac{sinx}{cosx} \end{equation*}[/tex]Box 2: Write as one expression
[tex]\begin{gathered} cotx-tanx=\frac{cosx}{s\imaginaryI nx}-\frac{s\imaginaryI nx}{cosx} \\ cotx-tanx=\frac{cosx(cosx)-sinx(sinx)}{sinxcosx} \\ cotx-tanx=\frac{cos^2x-sin^2x}{sinxcosx} \end{gathered}[/tex]The answer for the second box is
[tex]\frac{cos^{2}x-s\imaginaryI n^{2}x}{s\imaginaryI nxcosx}[/tex]Before the box 3, please note the identity
Note: Trigonometry I dentities
[tex]\begin{gathered} cos^2x-s\mathrm{i}n^2x=cos2x \\ 2sinxcosx=sin2x \end{gathered}[/tex]Box 3: Apply Double - Number Identities
[tex]\begin{gathered} cotx-tanx=\frac{cos^{2}x-s\imaginaryI n^{2}x}{s\imaginaryI nxcosx} \\ Applying\text{ the above trigonometry identities} \\ cotx-tanx=\frac{cos2x}{sinxcosx} \\ cotx-tanx=\frac{cos2x}{sinxcosx}\times\frac{2}{2} \\ cotx-tanx=\frac{2cos2x}{2sinxcosx} \\ cotx-tanx=\frac{2cos2x}{sin2x} \end{gathered}[/tex]Laverne created a painting with an area of 72 square inches and a length of 9 inches. They create a second painting with an area of 56 square inches. It has the same width as the first painting. What is the length of the second painting?
Answer: 7
Step-by-step explanation:
As our first step, we can divide 72 by 9 to discover the width of the first AND second painting. 72÷9=8
Now that we have discovered the width of the second painting, we will now solve the mathematical expression 56÷8.
Because 56 divided by 8 equals 7, the length of the second painting is 7 inches.
Question
Find the values for x and y .
Step-by-step explanation:
6x+3=75° ( being alternate angle )
6x = 72°
x=12
75+45+y= 180
y= 60°
Question 13 (3 points)
Intel's microprocessors have a 1.8% chance of malfunctioning. Determine the
probability that a random selected microprocessor from Intel will not malfunction.
Write the answer as a decimal.
EXPLANATION
The probability that Event A happening is the following:
[tex]P(A)[/tex]
The probability of Event A not happening is the following:
[tex]100-P(A)[/tex]Therefore, we have:
[tex]P(Malfunctioning)+P(Non\text{ Malfunctioning\rparen=100\%}[/tex]Plugging in the terms into the expression:
1.8 + P(Not malfunctioning) = 100%
Subtracting -1.8 to both sides:
[tex]P(Not\text{ malfunctioning\rparen=100-1.8}[/tex]Subtracting numbers:
[tex]P(Not\text{ malfunctioning\rparen=98.2}[/tex]In conclusion, the probability of not malfunctioning is 0.982
Factor the expression. 12y + 14
Answer:
2(6y+7)
Explanation:
To factor the expression:
[tex]12y+14[/tex]First, find the greatest whole number that divides 12 and 14.
The number = 2, therefore:
[tex]\begin{gathered} 12y+14=2\mleft(\frac{12y}{2}+\frac{14}{2}\mright) \\ =2(6y+7) \end{gathered}[/tex]There are 39 chocolates In a box call identically sheet dear 16 off filled with nuts 13 with caramel and 10 are solid chocolate you randomly select one piece eat it and then select a second piece find the probability of selecting to solid in a row
The probability of selecting two solid chocolates in a row is 0.0607 .
In the question ,
it is given that
there are total 39 chocolates in the box .
number of chocolates filled with nuts = 16
number of chocolates filled with caramel = 13
number of chocolates filled with solid = 10
Probability of selecting first chocolate as a solid is 10/39.
Now , there are 38 chocolates , with 9 solid chocolates ,
hence the probability of selecting second chocolate as a nut is = 9/38
So, the probability of selecting two solid chocolates in a row = 10/39×9/38
= 90/1482
= 0.0607
Therefore , the probability of selecting two solid chocolates in a row is 0.0607 .
The given question is incomplete , the complete question is
There are 39 chocolates in a box, all identically shaped. There 16 are filled with nuts, 13 with caramel, and 10 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting 2 solid chocolates in a row.
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Permutation/combination In how many ways can five people line up to get on a bus? (Hint: all 5 people are lining up)
We have to calculate in how many ways can five people line up to get on a bus.
This is a permutation of 5 in 5 without repetition, so it can be calculated as:
[tex]P(5,5)=5!=120[/tex]Answer: in 120 ways.
Solve for y Simplify your answer as much as possible Find by linear equation.
Given the equation:
[tex]-7=\frac{3y+7}{4}-\frac{9y-5}{2}[/tex]We will solve the equation to find y
Multiply the equation by 4 to eliminate the denominators
[tex]\begin{gathered} 4(-7)=4\cdot\frac{3y+7}{4}-4\cdot\frac{9y-5}{2} \\ \\ -28=(3y+7)-2(9y-5) \\ -28=3y+7-18y+10 \end{gathered}[/tex]Combine the like terms
[tex]\begin{gathered} -28=(3y-18y)+(7+10) \\ -28=-15y+17 \\ \end{gathered}[/tex]Subtract (17) to both sides
[tex]\begin{gathered} -28-17=-15y+17-17 \\ -45=-15y \end{gathered}[/tex]Divide both sides by (-15)
[tex]\begin{gathered} \frac{-45}{-15}=\frac{-15y}{-15} \\ \\ y=3 \end{gathered}[/tex]So, the answer will be y = 3
there are 14 square and 18 rectangles. what is the simplest ratio of squares to rectangles?
The simplest ratio of squares to rectangles can be obtained as follows:
There are 14 squares and 18 rectangles. The ratio of squares to rectangles is:
[tex]\frac{14}{18}=\frac{7}{9}[/tex]Then, the simplest ratio is 7/9 because 7 is a prime number and the ratio cannot be simplified any more. To obtain 7/9 we divided the numerator by 2 and the denominator also by 2.
I need help to know how to solve graphing a system of inequalities2x - 3y > -12x + y ≥ -2
Answer
2x - 3y > -12 (in red ink)
x + y ≥ -2 (in black ink)
The solution region is the region that the two shaded regions have in common.
Explanation
When plotting the graph of linear inequality equations, the first step is to first plot the graph of the straight line normally, using intercepts to generate two points on the linear graph.
If the inequality sign is (< or >), then the line drawn will be a broken line.
If the inequality sign is (≤ or ≥), then the line drawn is an unbroken one.
Step 1
For this question, we easily see that the first inequality will have a broken line and the second one will have an unbroken line.
To plot each of the lines, we will use intercepts to obtain the coordinates of two points on each line
Recall, we will first plot the lines like they are equations of a straight line.
To plot the graph
2x - 3y = -12
when x = 0,
2(0) - 3y = -12
-3y = -12
Divide both sides by -3
(-3y/-3) = (-12/-3)
y = 4
First point on the line is (0, 4)
when y = 0
2x - 3(0) = -12
2x = -12
Divide both sides by 2
(2x/2) = (-12/2)
x = -6
Second point on the line is (-6, 0)
For the second line,
To plot the graph,
x + y = -2
when x = 0
0 + y = -2
y = -2
First point on the line is (0, -2)
when y = 0
x + 0 = -2
x = -2
Second point on the line is (-2, 0)
So, for the plotting, we connect the two points for each of the lines.
Step 2
The shaded region now depends on whether the inequality sign is facing y or not.
If the inequality sign is facing y, it means numbers above the line plotted are the wanted region and the upper part of the graph is shaded.
If the inequality sign is not facing y, it means numbers below the line plotted are the wanted region and the lower part of the graph is shaded.
2x - 3y > -12
Can be rewritten as
-3y > -2x - 12
Divide through by -3 (this changes the inequality sign)
y < (2x/3) + 4
Here, we see that the inequality sign is not facing y, hence the numbers below the broken line plotted are the shaded region (in red ink)
x + y ≥ -2
We can rewrite this as
y ≥ -x - 2
Here, we see that the the inequality sign is facing y, hence, the numbers above the unbroken line plotted are the shaded region (in black ink)
The graph of this system of inequalities is presented above under 'Answer'
Hope this Helps!!!
Ronda ate 2/5 of the pie. Connor ate .375 of the pie. How much did they eatcombined? (Express your answer either as a fraction or decimal)
To convert 2/5 to a decimal;
[tex]undefined[/tex]I’ve been trying to figure out how to solve this and was wondering how to do this correctly!
SOLUTION
To solve this we will use the form for exponential growth to determine the formula to use.
Exponential growth has the form
[tex]\begin{gathered} P=P_0e^{rt} \\ P=\text{population after timer t} \\ P_0=\text{ initial population growth } \\ r=\text{ percent growth rate} \end{gathered}[/tex]Now the frogs tripple in population after 9 days. Initially they were 21. So in 9 days they become
[tex]21\times3=63\text{ frogs }[/tex]Applying the formula, we have
[tex]\begin{gathered} P=P_0e^{rt} \\ 63=21e^{9r} \\ 3=e^{9r} \\ \text{taking ln of both sides } \\ \ln 3=\ln e^{9r} \\ \ln 3=9r \\ r=\frac{\ln3}{9} \end{gathered}[/tex]The time for the frogs to get to 290 becomes
[tex]undefined[/tex]7 singles, 13 fives, 4 twenties, and 3 hundred dollar bills are all placed in a hat. If a player is to reach into the hat and randomly choose one bill, what is the fair price to play this game?
So,
The probability to obtain one bill of the four, is 1/4.
So, the expected value is:
[tex]\begin{gathered} \frac{1}{4}(7)+\frac{1}{4}(13)+\frac{1}{4}(4)+\frac{1}{4}(3) \\ \\ =6.75 \end{gathered}[/tex]Therefore, the fair price to play this game is 6.75.
A reflection across which line(s) carries the trapezoid onto itself?
If we reflect about x =2, it is a mirror image on each side ( left and right). There is no other line where we can have a mirror image on each side.
Robin is saving money to buy a 720$ phone. she has 105$ saved, and each week she adds 30$ to her savings. write an equation to find the number of weeks (w) until she has enough savings to buy the phone.
Let call w the number of weeks she has been saving.
Then, we can write the expression for her saving in function of the number of weeks as:
[tex]S(w)=105+30\cdot w[/tex]We now have to find the number of weeks it will take for her savings to reach $720.
We can find it by calculating w for S(w)=720:
[tex]\begin{gathered} S(w)-105+30w=720 \\ 105+30w=720 \\ 30w=720-105 \\ 30w=615 \\ w=\frac{615}{30} \\ w=20.5\approx21 \end{gathered}[/tex]Answer: it will take her 21 weeks to have enough savings.