we have the functions
[tex]\begin{gathered} f(x)=\frac{1}{x-2} \\ \\ g(x)=\sqrt{x+2} \end{gathered}[/tex]Find out f(g(x))
[tex]f\mleft(g\mleft(x\mright)\mright)=\frac{1}{\sqrt{x+2}-2}[/tex]Remember that
The radicand must be greater than or equal to zero and the denominator cannot be equal to zero
so
step 1
Solve the inequality
[tex]\begin{gathered} x+2\ge0 \\ x\operatorname{\ge}-2 \end{gathered}[/tex]the solution to the first inequality is the interval [-2, infinite)
step 2
Solve the equation
[tex]\begin{gathered} \sqrt{x+2}-2\ne0 \\ \sqrt{x+2}\operatorname{\ne}2 \\ therefore \\ x\operatorname{\ne}2 \end{gathered}[/tex]The domain is the interval
[–2, 2) ∪ (2, ∞)
The answer is the option CWhat is the volume of the following shape?
the multiple answers are A) 62,000 B) 32,000 C) 56,000 D) 52,000
I get 50,000.
I appreciate any help you can provide.
The volume of the shape is 59000.
Given that,
There is a picture in that we have a diagram.
We have to find the volume of the shape.
The shape we can divide into 2 parts.
Such as,
Cube and rectangular cube
That is by the measurements we can see
30 is the side of the cube then the length is 50+30 =80 and the width is 50-30=20.
The volume of the shape is Volume of the cube + Volume of rectangular cube.
The volume of the shape= a³+l×w×h
Here,
a is side of the cube that is 30
l is length of the rectangular cube is 80
w is width of the rectangular cube is 20
h is height of the rectangular cube is 20
The volume of the shape=30³+80×20×20
The volume of the shape=27000+32000
The volume of the shape=59000
Therefore, The volume of the shape is 59000.
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Write a system of equations and solve using elimination The sum of two numbers is 18. The difference between the two numbers is 2. What are the two numbers?
Assume that the 2 numbers are x and y, then
Since their sum is 18
That means add x and y, equate the sum by 18
[tex]x+y=18(1)[/tex]Since their difference is 2
That means subtract x and y, equate the difference by 2
[tex]x-y=2(2)[/tex]Add (1) and (2) to eliminate y
[tex]\begin{gathered} (x+x)+(y-y)=(18+2) \\ 2x+0=20 \\ 2x=20 \end{gathered}[/tex]Divide both sides by 2 to find x
[tex]\begin{gathered} \frac{2x}{2}=\frac{20}{2} \\ x=10 \end{gathered}[/tex]Substitute the value of x in (1) to find y
[tex]10+y=18[/tex]Subtract 10 from both sides
[tex]\begin{gathered} 10-10+y=18-10 \\ y=8 \end{gathered}[/tex]The 2 numbers are 10 and 8
Find the equation of the line
Use exact numbers
Answer:
y = (1)x +(-5)
Step-by-step explanation:
You want the slope-intercept equation of the line graphed with y-intercept -5 and x-intercept +5.
Intercept formThe intercept form of the equation for a line with x-intercept 'a' and y-intercept 'b' is ...
x/a +y/b = 1
Using the given intercept values, a=5, b=-5, the equation is ...
x/5 +y/(-5) = 1
Slope-intercept formThe desired form of the equation can be found by solving for y:
-x +y = -5 . . . . . . multiply by -5
y = x -5 . . . . . . . . add x
The numbers that go in the boxes are 1 and -5.
Hello there! can you help me on questions 3, 4, and 5 please? Thank you!
(3) A rate that is increasing means as the y variable increases, so does the x variable. This graph has its line rising up from left to right
(4) A graph with a rate that is decreasing would have the line sloping downwards from left to right (like the one provided in your question)
(5) A graph with a rate of zero means that, the y variable does not change, whereas the x variable keeps changing. This graph is a horizontal line (its parallel to the x-axis)
The rate is increasing
The rate is decreasing
The rate of change is zero
Which of the following has a graph that is an ellipse centered at (−2, 3)
Solution:
The vertex equation of an ellipse is;
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]Thus, given the center;
[tex](h,k)=(-2,3)[/tex]Then, the equation whose graph is an ellipse centered at (-2,3) is;
[tex]\frac{(x+2)^2}{12}+\frac{(y-3)^2}{18}=1[/tex]
6. For school uniforms, five shirts and three pairs of pants cost $113. 25. If a shirtcosts $3.75 less than a pair of pants, how much is a shirt and how much is a pairof pants?
Let x be the cost of 1 pair of pants, and y be the cost of one shirt, then since five shirts and three pairs of pants cost $113. 25 we can set the following equation:
[tex]5y+3x=113.25,[/tex]also, a shirt costs $3.75 less than a pair of pants, then:
[tex]y=x-3.75.[/tex]Substituting the second equation in the first one we get:
[tex]5(x-3.75)+3x=113.25.[/tex]Solving for x we get:
[tex]\begin{gathered} 5x-18.75+3x=113.25, \\ 8x=113.25+18.75, \\ x=\frac{33}{2}=16.5. \end{gathered}[/tex]Now, substituting x=16.5 in the second equation we get:
[tex]y=16.5-3.75=12.75.[/tex]Answer:
A shirt costs $12.75, and a pair of pants cost $16.5.
Hello! I need some help with this homework question, please? I just need help with C or D Q2
C) Considering that f(x)=x² we can write the composite function f(f(x)) y plugging into the x-variable the function f(x) itself:
[tex]\begin{gathered} f(f(x)) \\ f\mleft(x\mright)=x^2 \\ f(f(x))=(x^2)^2 \\ f(f(x))=x^4 \end{gathered}[/tex]Now, let's find the Domain. Considering that this is a polynomial function that has no restraints nor discontinuity we can write out the following:
[tex]\begin{gathered} The\: domain\: of\: f\circ f\: is\: all\: Real\: numbers \\ D=\: \mleft(-\infty\: ,\: \infty\: \mright) \end{gathered}[/tex]The remains of an ancient ball court include a rectangular playing alley with a perimeter of about 18m. The length of the alley is two times the width. Find the length and the width of the playing alley.The width is ? m and the length is ? m.
Given:
Perimeter = 18 m
The formula for the perimeter of a rectangle is:
[tex]P=2l+2w[/tex]Where:
l = lenght
w = width
In this case, we have that:
l = 2w
Therefore, we substitute the values in the formula:
[tex]\begin{gathered} P=2l+2w \\ 18=2(2w)+2w \end{gathered}[/tex]And solve for w:
[tex]\begin{gathered} 18=4w+2w \\ 18=6w \\ \frac{18}{6}=\frac{6w}{6} \\ w=3 \end{gathered}[/tex]For the length:
[tex]l=2w=2(3)=6[/tex]Answer:
The width is 3 m
The length is 6 m
Simplify 9y - 11 + 4y - 16y
The given expression is
9y - 11 + 4y - 16y
In order to simplify the expression, we would collect like terms. The like terms in this situation are
1) the terms containing y
2) the terms that don't contain y.
By collecting the like term, we would bring them together. It becomes
9y + 4y - 16y - 11
13y - 16y - 11
- 3y - 11
How do I write five hundred and eight hundredths as a decimal number.
start by writing the whole part, sine there us only hundreds we can write it as
[tex]500[/tex]then write the decimal part, we can see that there are no tenths in this expression so we start with a 0, then for the hundredths, we can put the number 8
[tex].08[/tex]The complete number should be
[tex]500.08[/tex]12. To prepare an aquarium for use, you can clean it with a saltwater solution. The amount of salt varies directly with the volume of the water. The solution has 2 teaspoons of aquarium salt for every gallon of water. a. How many teaspoons of aquarium salt are needed for 5 gallons of water? b. Write an equation that relates x gallons of water to y teaspoons of salt. c. Use the equation to find the number of gallons of water to use for 12 teaspoons of salt.
Given:
The solution has 2 teaspoons of aquarium salt for every gallon of water.
a.) How many teaspoons of aquarium salt are needed for 5 gallons of water?
- To be able to determine how many teaspoons of aquarium salt are needed, we will be using ratios and proportions.
[tex]\text{ 2 teaspoon of salt : 1 gallon of water = x : 5 gallons of water}[/tex]Where,
x = teaspoons of aquarium salt
We get,
[tex]\text{ 2 teaspoon of salt : 1 gallon of water = x : 5 gallons of water}[/tex][tex]\frac{2}{1}\text{ = }\frac{x}{5}[/tex][tex]\text{ (2)(5) = (x)(1)}[/tex][tex]\text{ 10 = x}[/tex]Therefore, you will be needing 10 teaspoons of aquarium salt for 5 gallons of water.
b.) Write an equation that relates x gallons of water to y teaspoons of salt.
Let,
x = gallons of water
y = teaspoons of salt
[tex]\text{ }\frac{2\text{ teaspoons of salt}}{1\text{ gallon}}\text{ = }\frac{y}{x}\text{ }\rightarrow\text{ }\frac{2}{1}\text{ = }\frac{y}{x}\text{ }\rightarrow\text{ 2 = }\frac{y}{x}[/tex][tex]\text{ 2x = y}[/tex]Therefore, the equation of the mixture will be 2x = y.
c.) Use the equation to find the number of gallons of water to use for 12 teaspoons of salt. Substitute, y = 12.
[tex]\text{ 2x = y}[/tex][tex]\text{ 2x = 12}[/tex][tex]\text{ }\frac{2x}{2}\text{ = }\frac{12}{2}[/tex][tex]\text{ x = 6}[/tex]Therefore, you will be needing 6 gallons of water for 12 teaspoons of salt.
Convert the following mixed number to an improper fraction: 28 1 / 8 What is the numerator of this improper fraction? State the answer without reducingAll negatives must be included in your final answer:ie. if the question asks for the whole number portion, numerator, or denominator of the answer and it has a negative, you must include that negative.Be sure to reduce all fractions fully and convert improper fractions to mixed numbers before stating your final answer
Given the mixed number:
[tex]28\frac{1}{8}[/tex]Let's convert the mixed number to an improper fraction.
To convert a mixed number to improper fraction, multiply the denominator by the whole number, then add the result to the numerator.
Thus, we have:
[tex]\begin{gathered} 28\times8=224 \\ \Longrightarrow224+1=225 \end{gathered}[/tex]Therefore, the improer fraction is:
[tex]\frac{225}{8}[/tex]The numerator is the number at the top of the fraction.
Therefore, the numerator of this improper fraction is = 225
ANSWER:
[tex]\frac{225}{8}[/tex]Numerator = 225
Answer: To convert a mixed number to improper fraction multiply the whole number and the denominator and then add the numerator. This gives the numerator of the improper fraction. The denominator is the same as the denominator of the given mixed number.
Step-by-step explanation:
[tex]28\frac{1}{8}[/tex] = [28x8+1] / 8 = [tex]\frac{224+1}{8}[/tex] = [tex]\frac{225}{8}[/tex]
The numerator of [tex]\frac{225}{8}[/tex] is 225
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Jackie cut 3-yards of ribbon into 6 equal lengths of ribbon. What is the length of each ribbon in yards? A. C. 1 2 B. 18 1 colo D.2 ОА OB Ос OD (
The ribbon is 3 yards. He cuts the ribbon into 6 equal length. The length of each ribbon can be calculated below
total length of the ribbon = 3 yards
[tex]\begin{gathered} \text{each length of the cut ribbon=}\frac{3}{6} \\ \text{each length of the cut ribbon}=\frac{1}{2}\text{ yards} \end{gathered}[/tex]Write an addition fact that corresponds to the following sentence, and then answer the questionOn three consecutive passes, a football team gains 10 yards, loses 19 yards, and gains 21 yards. What number represents the total net yardage?The addition fact that corresponds to the sentence is_____The total net yardage is yards______
Given:
On three consecutive passes, a football team gains 10 yards, loses 19 yards, and gains 21 yards.
To find:
An addition fact that corresponds to the given sentence and the number represents the total net yardage.
Solution:
It is given that first football gains 10 yards then loses 19 yards and then again gains 21 yards.
The gaining is shown by a positive sign and the loss is shown by a negative sign.
So, the addition fact is given below:
[tex]+10-19+21[/tex]Now, the resultant of the above expression is the total net yardage.
[tex]\begin{gathered} +10-19+21=+31-19 \\ =+12 \end{gathered}[/tex]Thus, the total net yardage is 12 yards.
Use the appropriate form of the percentage formula.30% of what number is 21?
If 30% of a number x is equal to 21, then x is to 100% as 21 is to 30%:
[tex]\frac{x}{100}=\frac{21}{30}[/tex]Solve for x:
[tex]x=\frac{21}{30}\times100=70[/tex]Then, 30% of 70 is 21.
Therefore, the answer is 70.
At what point will the lines x= -21 - 2y and x = -39 - 4y intersect? (i need the answer)
ANSWER
The lines intersect at (-3, -9)
EXPLANATION
The point where the lines intersect is the solution to the system of equations
[tex]\begin{cases}x=-21-2y \\ x=-39-4y\end{cases}[/tex]We can solve it using the elimination method, which consists in subtracting one equation from the other in order to eliminate one of the variables and obtain only one equation with only one variable:
[tex]\begin{gathered} x=-21-2y \\ - \\ x=-39-4y \\ \text{ ------------------------------} \\ (x-x)=(-21-2y)-(-39-4y) \end{gathered}[/tex]Solving for y:
[tex]\begin{gathered} 0=-21+39-2y+4y \\ 0=18+2y \\ 2y=-18 \\ y=-9 \end{gathered}[/tex]Now we have to replace y = -9 into one of the equations and solve for x:
[tex]\begin{gathered} x=-21-2y \\ x=-21-2(-9) \\ x=-21+18 \\ x=-3 \end{gathered}[/tex]The lines intersect at (-3, -9)
Simplify. In the form of a paragraph, explain in complete sentences the steps necessary to simplify the expression andinclude the final answer in your explanation. Complete your work in the space provided or upload a file that can displaymath symbols if your work requires it.
1. When dividing with exponents, the exponent of a variable in the denominator is subtracted from the exponent in the numerator for the same variable. Then, first step to simplify is subtract the exponents of x and y in the fraction in parentheses:
[tex]\begin{gathered} =(x^{3-1}y^{1-2})^{-2} \\ =(x^2y^{-1})^{-2} \end{gathered}[/tex]2. To remove the parentheses you multiply each exponent in the parentheses by the exponent out of the parentheses:
[tex]\begin{gathered} =x^{2\cdot(-2)}y^{(-1)\cdot(-2)} \\ \\ =x^{-4}y^2 \end{gathered}[/tex]3. When you have a negative exponent (as the x powered to -4) you divide 1 in to the term with negative exponent (after you divide the exponent turns into a positive exponent):
[tex]=\frac{1}{x^4}\cdot y^2[/tex]4. Then, the given expression simplified is:
[tex](\frac{x^3y}{xy^2})^{-2}=\frac{y^2}{x^4}[/tex]PLEASE HELP!!!!! I WILL GIVE BRAINLIEST!!!!
Evaluate the expression . Write your answer as a simplified mixed number, or as a decimal. Show your work
Answer:
[tex]-3\frac{15}{16}[/tex]
Step-by-step explanation:
Given:
[tex](-\frac{1}{4}+2.875)\div(-\frac{2}{3})[/tex]
First, convert 2.875 to an improper fraction:
[tex]2.875=2\frac{875}{1000}=2\frac{7}{8}=\frac{23}{8}[/tex]
The expression becomes:
[tex](-\frac{1}{4}+\frac{23}{8})\div(-\frac{2}{3})[/tex]
Then, add the terms in the parentheses:
[tex]-\frac{1}{4}+\frac{23}{8}=-\frac{2}{8}+\frac{23}{8}=\frac{21}{8}[/tex]
The new expression is:
[tex]\frac{21}{8}\div-\frac{2}{3}[/tex]
Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. Therefore:
[tex]\frac{21}{8}\times-\frac{3}{2}[/tex]
Next, multiply the numerators and denominators:
[tex]-\frac{63}{16}[/tex]
Finally, simplify to a mixed fraction:
[tex]-3\frac{15}{16}[/tex]
Harry wants to buy a motorbike that costs £600.He saves 30% of his wages each month.Each month Harry earns £400.Calculate how many months it will take Harry to save enough money?
First, let's calculate how much money he saves each month, by calculating 30% of £400:
[tex]\frac{30}{100}\cdot400=30\cdot4=120[/tex]He saves £120 each month.
So, to reach the value of £600, the number of months needed is:
[tex]\frac{600}{120}=5[/tex]Therefore it will take 5 months to save enough money.
At the grocery store, 3 apples cost $0.65. What is the cost of 8 apples? 7th honors it is due today help
Answer:
$1.76 for 8 apples
Step-by-step explanation:
0.65 / 3 = 0.22
0.22 * 8 = 1.76
The water level of a tank every minute since it began filling is indicated by segments A,B,and C on the graph
SOLUTION
From the graph
The slope of line A is
[tex]m=\frac{60-20}{2-0}=20[/tex]The slope of line B is
[tex]n=\frac{80-60}{6-2}=5[/tex]The slope of line C is
[tex]p=\frac{110-80}{9-6}=10[/tex]The least segment is the segment with the least slope.
The required arrangement is
[tex]B,C,A[/tex]assume that y varies inversely with x. If y = -4 when x = 1/2 , find x when y=2
it is given that x and y have inverse relation
so K = xy
put y = -4 and x = 1/2
[tex]\begin{gathered} k=\frac{1}{2}\times-4 \\ k=-2 \end{gathered}[/tex]now
y = 2
then'
[tex]\begin{gathered} -2=x\times2 \\ x=\frac{-2}{2} \\ x=-1 \end{gathered}[/tex]so the value of x = -1
[tex]\begin{gathered} x\infty\frac{1}{y} \\ x=\frac{K}{y} \\ K=xy \end{gathered}[/tex]how much time will it take for a fire to reach this machine size
Given the equation:
[tex]y=4(1.8){}^t[/tex]You know that "y" represents the number of acres of land, and "t" represents the number of minutes the fire has raged.
In order to find the time (in minutes) it will take for a fire to reach 160 acres, you need to substitute this value of "y" into the equation:
[tex]y=160[/tex]And then solve for "t":
[tex]160=4(1.8){}^t[/tex]Follow these steps in order to solve for "t":
- Divide both sides of the equation by 4:
[tex]\begin{gathered} \frac{160}{4}=\frac{4(1.8){}^t}{4} \\ \\ 40=(1.8)^t \end{gathered}[/tex]- Take the logarithm from both sides:
[tex]\begin{gathered} log(40)=log(1.8)^t \\ \\ log(40)=t\cdot log(1.8) \end{gathered}[/tex]- Divide both sides by the logarithm on the right side of the equation:
[tex]\begin{gathered} \frac{log(40)}{log(1.8)})=\frac{t\cdot log(1.8)}{log(1.8)} \\ \\ t\approx6.28 \end{gathered}[/tex]Hence, the answer is:
[tex]6.28\text{ minutes \lparen approximately\rparen}[/tex]The quadratic equation y = - 16t² +40t+2 represents the height of aprojectile, y, in feet, at a particular time, t, in seconds.For what interval or intervals of time will the projectile be above 18 feet?
The given equation is:
[tex]y=-16t^2+40t+2[/tex]It is required to find which interval or intervals of time will the projectile be above 18 feet.
To do this, solve the inequality:
[tex]\begin{gathered} y>18 \\ \Rightarrow-16t^2+40t+2>18 \end{gathered}[/tex]First, find the critical points of the inequality by solving the equation:
[tex]\begin{gathered} −16t^2+40t+2=18 \\ \text{ Subtract 18 from both sides:} \\ \Rightarrow-16t^2+40t+2-18=18-18 \\ \Rightarrow−16t^2+40t-16=0 \\ \text{ Factor the left-hand side of the equation:} \\ \Rightarrow−8\left(2t−1\right)\left(t−2\right)=0 \\ \text{ Equate the factors to 0 to find the t-values:} \\ \Rightarrow(2t-1)=0\text{ or }(t-2)=0 \\ \Rightarrow2t=1\text{ or }t=2 \\ \Rightarrow t=\frac{1}{2}=0.5\text{ or }t=2 \end{gathered}[/tex]The possible interval of solutions are:
[tex]t<0.5,\;0.52[/tex]Use test values in the intervals to check which interval whose set of values satisfies the given inequality.
The only interval that satisfies it is 0.5.
Hence, the answer is between 0.5 second and 2 seconds.
The answer is option (c).-4(-7x-6) help me please
multiply each term by -4 and sum
[tex]\begin{gathered} (-4\times-7x)+(-4\times-6) \\ =28x+24 \\ \end{gathered}[/tex]Triangle ABC, with vertices A(-2, 4), B(0, -5), and C(-3, -8) was dilated to form triangleA'B'C' with vertices A(-1.8, 3.6), B(0, -4.5), and C(-2.7, -7.2). What rule represents thedilation applied?(x, y) = (2/3x, 2/3y)(x, y) - (0.9x, 0.9y)(x, y) - (-X, -Y)(x, y) - (1.2x, 1.25y)
A = (-2,4)
B= (0,-5)
C = (-3,-8)
A'= (-1.8, 3.6 )
B'= (0, -4.5 )
C' = (-2.7, -7.2)
Try each option:
(x, y) = (2/3x, 2/3y)
A = (-2,4) = (2/3*-2 , 2/3*4) = (-4/3, 8/3) NOt equal to A'
(x, y) - (0.9x, 0.9y) =
A = (-2,4) = (0.9*-2, 0.9*4) = (-1.8, 3.6) = A' YES
B = (0,-5) = (0.9*0, 0.9*-5) = (0, -4.5) = B'
C= (-3,-8) = (0.9*-3, 0.9*-8) = ( -2.7, -7.2)= C'
Correct option : (x, y) - (0.9x, 0.9y)
Hi, can you help me answer this question please, thank you
From the given z-value z=1.841, we can find the corresponding P-value by means of a z-table:
Then, by rounding to 4 decimal places, the p-value is 0.9672
choose the image that corresponds to Figure 1 after a reflection over the x-axis and a translation of one unit left
Solution
For this case the solution would be given by:
There are 150 people at an International Medical Conference. 40 are Africans, 70 are women and 110 are doctors. 12 of the women are Africans, 46 of the doctors are women and 31 of the Africans are doctors. If 5 of the African men are not doctors: a how many of the African women are doctors b how many of the men are neither African nor doctors?
Draw a Venn Diagram to visualize the situation:
Notice that the numbers in purple correspond to the total amount of people in the regions P and O, P and Q, P and S, respectively.
Numbers in red correspond to the number of people in each region.
Then, we should find the correct numbers for the regions O, P, Q and S, as well as M, N, R and T so that the total number of people in each group matches the labels.
Let x represent the amount of people in the region P.
Since the total amount of people in regions O and P must be 31, then the amount of people in O must be 31-x.
Similarly, the amount of people in region Q must be 12-x.
Since the amount of people in the region N (African men who are not doctors) is 5 and the total amount of African people is 40, then, we can write down an equation for x, where the sum of the amount of people in regions N, O, P and Q is 40:
[tex]5+(31-x)+x+(12-x)=40[/tex]Solve for x:
[tex]\begin{gathered} \Rightarrow5+31-x+x+12-x=40 \\ \Rightarrow48-x=40 \\ \therefore x=8 \end{gathered}[/tex]Notice that the number of people in region P, which is 8, corresponds to amount of African women who are doctors. Therefore, the answer for part a) is: 8.
To find how many of the men are neither African or doctors, find first the correct amount of people in regions R and T. This can be done by taking into account the total amount of people who are doctors and the total amount of people who are women.
The number of people in regions O, P, R and S must be equal to 110. For that to happen, the number of people in region R must be 41.
The amount of people in region T can be found similarly, and it is equal to 20.
Once the total amount of people in regions N, O, P, Q, R, S and T is known, we can deduce the number of people that must be in region M taking into account that the total amount of people in the conference is 150.
Substract the total amount of people in regions N to S (which is 139) from 150, in order to find the total amount of people in region M:
[tex]150-139=11[/tex]The region M corresponds to men who are neither African or doctors. Therefore, the answer to part b) is 11.
determine the value of d22, if possible. if the element indicated is not in the matrix, state “none”.
Answer:
[tex]5[/tex]Explanation:
Here, we want to determine the given element
From what we have, it is stated as d22
What this simply means is that we are looking at the element in row 2, column 2
The element here is:
[tex]5[/tex]