we have the following:
[tex]\begin{gathered} x=\frac{30}{2.5} \\ x=12 \end{gathered}[/tex]therefore the number of galllons is 12
5.73×10^6 scientific notation or standard form
The number 5.73 10^6 is shown in scientific notation.
Its standard form is: 5,730,000 (five million 7 hundred thirty thousand)
The number 600 can be written in scientific notation by using the digit "6" followed by a product by 100: 6 * 100 and now writing the 100 in powers of ten:
100 = 10 * 10 = 10^2
then the scientific notation formof 600 is:
6 10^2 (6 times 10 to the power 2)
The number 0.24 is the same as the number 24 divided by 100 so notice that there is DIVISION by powers of ten now, and such division becomes a "negative" power of the base 10.
The way to quickly write 0.24 in scientific notation is:
1) count how many spaces you have to move the decimal point to the right in order to get to a number between 1 and something smaller than 10. In our case to get to 2.4.SO the decimal point has to move ONE space to the right. Now,that number of spaces you move to the right is going to becoma the exponent of the base 10:
2.4 10 ^(-1) (remember that since this is a division, the power is NEGATIVE.
The number 4.07369 is the same written in scientific notation and in standard form, because the number shown is in between 1 and something smaller than 10.
Your teacher may want tho to have yu write the exponent of 10 (you write it in this case as a zero):
4.07369 = 4.07369 10^0
Recall that 10^0 is 1, so there is no actual change to the number by multiplying by it.
A bank offers a CD that pays a simple interest rate of 2.5%. How much must you put in this CD now in order to have $4,000 for a home-entertainmentcenter in 2 years.
The formula to calculate Simple Interest is given as
[tex]I=\frac{\text{PRT}}{100}[/tex]The question provides the following parameters:
[tex]\begin{gathered} R=2.5 \\ T=2 \end{gathered}[/tex]If the amount to be had now is $4000, which is inclusive of the interest to be had over the period, this means that
[tex]P+I=4000[/tex]If we substitute the value for I, we have a new equation, such that
[tex]\begin{gathered} P+\frac{\text{PRT}}{100}=4000 \\ \therefore \\ P(1+\frac{RT}{100})=4000 \end{gathered}[/tex]Substituting the values into the equation, we can solve for P as
[tex]\begin{gathered} P(1+\frac{2.5\times2}{100})=4000_{} \\ P(1+0.05)=4000 \\ 1.05P=4000 \\ P=\frac{4000}{1.05} \\ P=3809.52 \end{gathered}[/tex]The answer is $3,809.52
here is some formula of understandable things. [tex] {14471}^{2852} \times 1 + 2 - {1666}^{3} \div 145663 \times \frac{5}{3} + \sqrt{86} \tan(5) + \pi0.14 = [/tex]
here is the session with equation in question
A rectangular garden is 42ft wide and 72 ft long.A blueprint is created using a scale of 1in:6ft.Find the length and width of the blueprints and do not include units in yours answers.A. Identify the length on the blueprint.B. Identify the width on the blueprint.
EXPLANATION
As we already know, scale factor is the number by wich all the components of an object are multiplied in order to create a proportional enlargement or reduction.
First, we need to turn 6ft into inches units in order to have same magnitudes.
1 ft = 12 inches
So, the relationship is now 1in: 12 in
Scale factor = blueprint size / garden size
Isolating the blueprint size:
BluePrint size = Scale Factor * Garden Size
So, replacing terms:
----> The width will be 42* 1/12 = 3.5
----> The length will be 72 * 1/12 = 6
Answers:
A. The length of the blueprint is 6
B. The width of the blueprint is 3.5
after a raise Alex salary increased from 30000 anually to 31590 find the percent
intial value = 30,000
final value = 31,590
30,00 ( 1 + x) = 31,590
Solve for x ( increase in decimal form)
30,000+ 30,000x = 31,590
30,000x = 31,590-30,000
30,000x =1,590
x= 1,590/30,000
x= 0.053
Multiply by 100
0.053 x 100= 5.3%
8 singles, 10 fives, 2 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?
The total number of bills are 23.
The probability to get single = 8/23
The probability to get five = 10/23
The probability to get twenty = 2/23
The probability to get a hundred = 3/23
So, the fair price to play this game is calculated below:
[tex]\begin{gathered} \text{fair price}=1\times\frac{8}{23}+5\times\frac{10}{23}+20\times\frac{2}{23}+100\times\frac{3}{23} \\ =\frac{8}{23}+\frac{50}{23}+\frac{40}{23}+\frac{300}{23} \\ =\frac{8+50+40+300}{23} \\ =\frac{398}{23} \\ =17.30 \end{gathered}[/tex]Thus, the fair price to play this game is $17.30
evaluate the expression 3y+6÷2x
3y+6÷ 2x = 1/ 2x ( 3y + 6 )
Kylee manages a small theme park and she has been analyzing the attendance data. Kylee finds that the number of visitors increases exponentially as the temperature increases, and this situation is represented by the function f(x) = 4x. Kylee also finds a linear equation that models the number of people who leave the park early depending on the change in temperature, and it is represented by g(x) = −x + 5. The graph of the two functions is below. Find the solution to the two functions and explain what the solution represents.
Thus, the solution is (1,4) and that represents the number in which the number of visitors incoming when the temperature is increasing matches the number of visitors leaving early when the temperature is decreasing.
1) The solution to both functions is that point that is located at the intersection of the curve and the line.
2) So, let's solve this system of equations:
[tex]\begin{gathered} \begin{matrix}y=4^x\end{matrix} \\ y=-x+5 \\ \end{gathered}[/tex]Note that we can apply the Substitution Method:
[tex]\begin{gathered} 4^x=-x+5 \\ \ln4^x=\ln_(-x+5) \\ x\ln(4)=\ln_(-x+5) \\ x=\frac{\ln(-x+5)}{\ln(4)} \\ \frac{\ln \left(-x+5\right)}{\ln \left(4\right)}=x \\ \frac{\ln \left(-x+5\right)}{\ln \left(4\right)}\ln \left(4\right)=x\ln \left(4\right) \\ \ln \left(-x+5\right)=x\ln \left(4\right) \\ \ln \left(-x+5\right)=2\ln \left(2\right)x \\ e^{\ln \left(-x+5\right)}=e^{2\ln \left(2\right)x} \\ -x+5=4^x \\ -(1)+5=4^1 \\ 4=4 \\ x=1 \end{gathered}[/tex]With the quantity of x=1 we can plug it into the second formula:
3)
[tex]\begin{gathered} y=-x+5 \\ y=-1+5 \\ y=4 \end{gathered}[/tex]4) Thus, the solution is (1,4) and that represents the number in which the number of visitors incoming when the temperature is increasing matches the number of visitors leaving.
Given: Circle HHKJK is best described as aof Circle H.JK is best described as a
A chord of a circle is a straight line segment whose endpoints both lie on a circular arc.
In geometry, a secant is a line that intersects a curve at a minimum of two distinct points.
Answer:
A chord of a circle is a straight line segment whose endpoints both lie on a circular arc.
In geometry, a secant is a line that intersects a curve at a minimum of two distinct points.
Step-by-step explanation:
I’m struggling with my homework assignment, can anyone help me?
We can find the inverse function writing f(x) as y in the original function and then changing x with y and isolating y again, so
[tex]y=\sqrt[]{x}-5[/tex]Changing y with x we have
[tex]x=\sqrt[]{y}-5[/tex]Now we must get y on one side of the equation again, then
[tex]\begin{gathered} x=\sqrt[]{y}-5 \\ x+5=\sqrt[]{y} \\ (\sqrt[]{y})^2=(x+5)^2 \\ y=(x+5)^2 \end{gathered}[/tex]The domain of the inverse function in the image of the original function f, the image of f is x ≥ -5, then the domain of the inverse of will be x ≥ -5, so our answer is
[tex]f^{-1}(x)=(x+5)^2,x\ge-5[/tex]A county fair sells adult admission passes, child admission passes, and ride tickets. One family paid $29 for two adult passes,three child passes, and nine ride tickets. Another family paid $19 for one adult pass, two child passes, and eight ride tickets. A third family paid $51 for three adult passes,five child passes, and twenty one ride tickets. Find the individual costs of an adult pass,a child pass, and a ride ticket? Show all work
x= adult passes
y= child passes
z=ride tickets
the first family:
[tex]2x+3y+9z=29\text{ (1)}[/tex]the sencond family
[tex]x+2y+8z=19\text{ (2)}[/tex]third family
[tex]3x+5y+21z=51\text{ (3)}[/tex]now we have the 3 equations, and we can solve x, y and z
for the equation of the second family we have:
[tex]x=-2y-8z+19\text{ (4)}[/tex]reeplace the new equation(4) in (1), we have:
[tex]2(-2y-8z+19)+3y+9z=29[/tex][tex]-4y-16z+38+3y+9z=29[/tex][tex]y+7z=9\text{ (5)}[/tex]reeplace (4) in (3)
[tex]3(-2y-8z+19)+5y+21z=51[/tex][tex]-6y-24z+57+5y+21z=51[/tex][tex]-y-3z=-6[/tex][tex]y+3z=6\text{ (6)}[/tex]with 5 and 6, we have a 2x2 equation
that we can solve easier
solving 5 and 6, we have:
[tex]\begin{gathered} y+7z=9 \\ y=9-7z\text{ (7)} \end{gathered}[/tex]reeplace 7 in 6
[tex]\begin{gathered} 9-7z+3z=6 \\ 4z=3 \\ z=\frac{3}{4}=0.75 \end{gathered}[/tex]now we find y, reeplace z in (7)
[tex]\begin{gathered} y=9-7(0.75) \\ y=9-5.25 \\ y=3.75 \end{gathered}[/tex]and finally we can find x, reeplacing y and z in (4)
[tex]\begin{gathered} x=-2y-8z+19 \\ x=-2(3.75)-8(0.75)+19 \\ x=-7.5-6+19 \\ x=5.5 \end{gathered}[/tex]the length of the longer leg of a right triangle is 3 ft more than three times the length of the shorter leg. the length of the hypotenuse is 4 ft more than three times the length of the shorter leg. find the side lengths of the triangle.
with the pythagorean theorem
[tex]\begin{gathered} (4+3x)^2=x^2+(3+3x)^2 \\ 16+24x+9x^2=x^2+9+18x+9x^2 \\ 16+24x+9x^2=10x^2+18x+9 \\ 16+24x+9x^2-9=10x^2+18x+9-9 \\ 9x^2+24x+7=10x^2+18x \\ 9x^2+24x+7-18x=10x^2+18x-18x \\ 9x^2+6x+7=10x^2 \\ 9x^2+6x+7-10x^2=10x^2-10x^2 \\ -x^2+6x+7=0 \end{gathered}[/tex]using the formula of the quadratic equation
[tex]\begin{gathered} x_{1,\: 2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ x1=\frac{-6+\sqrt{6^2-4\left(-1\right)\cdot\:7}}{2\left(-1\right)}=-1 \\ x2=\frac{-6-\sqrt{6^2-4\left(-1\right)\cdot\:7}}{2\left(-1\right)}=7 \end{gathered}[/tex]the length cannot be negative, therefore x=7
length of the shorter leg is: 7ft
length of the longer leg is: 3+3(7)= 24ft
length of the hypotenuse is: 4+3(7)= 25ft
Sophie has eamed $3500 working at the movie theater decides to put her money in the bank in an account that has a 7.05% interest rate that is compounded continuously write an equation to model this!
Step 1. The information we have is.
The initial amount of the investment which is called the principal P is:
[tex]P=3500[/tex]The interest rate is 7.05%, this will be r:
[tex]r=7.05\text{ percent}[/tex]We will need to represent the interest rate as a decimal number, for that, we divide by 100:
[tex]\begin{gathered} r=\frac{7.05}{100} \\ \downarrow \\ r=0.0705 \end{gathered}[/tex]As additional variables, we will have:
[tex]\begin{gathered} A\longrightarrow\text{Total amount} \\ t\longrightarrow\text{time of the investment} \end{gathered}[/tex]Step 2. Use the Continuous compounding formula:
[tex]A=Pe^{rt}[/tex]where A is the amount including interest, P is the principal amount of the investment, r is the interest rate, and t in years.
Also, e is a constant which is equal to:
[tex]e\approx2.783[/tex]But we will only represent it as e.
Step 3. Substitute P and r into the continuous compounding formula:
[tex]\boxed{A=3500e^{0.0705\times t}}[/tex]That is the equation that models the situation.
Answer:
[tex]\boxed{A=3500e^{0.0705\times t}}[/tex]Which inequality represents the phrase, the quotient of w and four is at least 3.
The inequality that represents the phrase is:
[tex]\frac{w}{4}\ge3[/tex]Unsure of this one, I need it explained with the answer
ANSWER
k = 1 or 21
STEP-BY-STEP EXPLANATION:
According to the question, we were given the below trigonometric function
[tex]\sec ^2x\text{ - 22tanx + 20 = 0}[/tex]Recall that, we have trigonometric identity which is written below
[tex]\sin ^2\theta+cos^2\theta\text{ = 1}[/tex][tex]\text{Divide through by }\cos ^2\theta[/tex][tex]\begin{gathered} \frac{\sin^2\theta}{\cos^2\theta}\text{ + }\frac{cos^2\theta}{\cos^2\theta}\text{ =}\frac{1}{\cos ^2\theta} \\ \tan ^2\theta+1=sec^2\theta \\ \text{Let x = }\theta \\ \tan ^2x+1=sec^2x \end{gathered}[/tex]The next thing is to rewrite the equation
[tex]\begin{gathered} \text{ since sec}^2x=tan^2x\text{ + 1} \\ \text{Hence,} \\ \tan ^2x\text{ + 1 - 22tanx + 20 = 0} \\ \text{Let k = tanx} \\ k^2\text{ + 1 -22k + 20 = 0} \\ \text{Collect the like terms} \\ k^2\text{ - 22k + 21 = 0} \end{gathered}[/tex]The next thing is to find the value of P by factorizing the above equation.
Recall that, the standard form of the quadratic function is given as
[tex]ax^2\text{ + bx + c = 0}[/tex]Let
a = 1
b = -22
c = 21
The next thing is to find the value of ac
[tex]\begin{gathered} ac\text{ = 1 }\cdot\text{ 2}1 \\ ac\text{ = 2}1 \end{gathered}[/tex][tex]\begin{gathered} k^2\text{ - k -21k + 21 =0} \\ k(k\text{ -1) -21(}k\text{- 1) = 0} \\ (k\text{ -1) (k -21) = 0} \\ k\text{ -1 = 0 or =k - 21 = 0} \\ k\text{ = 1 or k = 22} \end{gathered}[/tex]Hence, the value of k is 1 or 21
what is a quadrilateral with 4 congruent sides and 4 right angles called?
what is a quadrilateral with 4 congruent sides and 4 right angles called .......................
a Parallelogram with four congruent sides and four right angles.
I need help on question 13 for a b and c
b) We have to calculate the probability that a group of 25 men exceeds the average allowed weight per passenger.
As the water taxi has a load limit of 3500 lb, the maximum average weight per passenger is 3500/25 = 140 lb.
Then, we can calculate the probability that the mean of a sample of size n = 25 is greater than 140 lb.
The population distribution from where the sample is taken has a mean of 189 lb and a standard deviation of 39 lb.
We can calculate the z-score for M = 140 lb for this sample as:z
[tex]z=\frac{M-\mu}{\sigma\/\sqrt{n}}=\frac{140-189}{39\/\sqrt{25}}=\frac{-49}{39\/5}\approx-6.28[/tex]Then, the probabilitty can be expressed as:
[tex]P(M>140)=P(Z>-6.28)\approx1[/tex]It is almost certain that the sample mean will be greater than 140 lb.
c) We now have to calculate the probability that a sample of size n = 20 has a mean that is greater than 175 lb, the new load limit per passenger.
We can repeat the procedure calculating the z-score with this new values (M = 175 and n = 20):
[tex]z=\frac{M-\mu}{\sigma\/\sqrt{n}}=\frac{175-189}{39\/\sqrt{20}}\approx\frac{-14}{8.721}\approx-1.6[/tex]Then, we can look up the probability for the standard normal distribution when z = -1.6 and obtain:
We can express this as:
[tex]P(M>175)=P(z>-1.6)=0.9452[/tex]d) As the probabilty of exceeding the load limit per passenger is too high, we can consider that 20 passengers is still not safe enough.
Answer:
b) P(M > 140) = 1
c) P(M > 175) = 0.9452
d) Not safe
Find the length of the guy wire. If necessary, round to the nearest tenth foot.
We are given a diagram showing a pole with a guy wire attached to the top of it and anchored into the ground.
From the base of the pole to the bottom end of the guy tower is given as a 20-feet distance. The pole itself is 24 feet tall. The guy wire from the top of the pole to the ground forms the hypotenuse of what we can describe as a right angled triangle.
We can now use the Pythagoras' theorem to solve for the missing side (hypotenuse).
The theorem states;
[tex]c^2=a^2+b^2[/tex]Where the variables are;
[tex]\begin{gathered} c=\text{hypotenuse} \\ a,b=\text{other sides} \end{gathered}[/tex]We can now substitute the values given;
[tex]c^2=24^2+20^2[/tex][tex]c^2=576+400[/tex][tex]c^2=976[/tex]Take the square root of both sides;
[tex]\sqrt[]{c^2}=\sqrt[]{976}[/tex][tex]c=31.240998\ldots[/tex]Rounded to the nearest tenth of a foot, the length of the guy wire is;
ANSWER:
Length = 31.2 ft
The second option is the correct answer.
Solve for the height h in this right triangle. Show all steps and round your answer to thenearest hundredth.h39°875 feet
Step 1
Name all sides
Side facing given angle is the opposite = h
Side facing right angleis the hypotanuse
The other side is the adjacent = 875 feet
[tex]\begin{gathered} \tan \theta\text{ = }\frac{opposite}{\text{adjacent}} \\ \theta\text{ = 39} \end{gathered}[/tex][tex]\begin{gathered} \tan 39\text{ = }\frac{h}{875} \\ 0.809\text{ = }\frac{h}{875} \\ \text{Cross multiple} \\ h\text{ = 0.809 x 875} \\ h\text{ = 708.561} \\ h\text{ = 708.56 } \end{gathered}[/tex]"10 more than one-fourteenth of some number, w" can be expressed algebraically as
To express the statement algebraically, we will break down the sentence into phrases or words we can interprete numerically:
the unknown number = w
one fourteenth = 1/14
one fourteenth of some number w:
[tex]\begin{gathered} =\frac{1}{14}\times w \\ =\text{ }\frac{w}{14} \end{gathered}[/tex]10 more: it means we will be adding 10 to the algebraic expression after
10 more than one-fourteenth of some number w will be:
[tex]=10\text{ + }\frac{w}{14}[/tex]Can I get help with B? I just need to find the standard deviation
Given the set of data:
11, 7, 14, 2, 8, 13, 3, 6, 10, 3, 8, 4, 8, 4, 7
Let's find the standard deviation.
To find the standard deviation, apply the formula:
[tex]s=\frac{\sqrt{\Sigma(x-\mu)^2}}{n-1}[/tex]Where:
x is the data
u is the mean
n is the number of data = 15
To find the mean, we have:
[tex]\begin{gathered} mean=\frac{11+7+14+2+8+13+3+6+10+3+8+4+8+4+7}{15} \\ \\ mean=\frac{108}{15} \\ \\ mean=7.2 \end{gathered}[/tex]Hence, to find the standard deviation, we have:
[tex]\begin{gathered} s=\sqrt{\frac{(11-7.2)^2+(7-7.2)^2+(14-7.2)^2+(2-7.2)^2+(8-7.2)^2+(13-7.2)^2+(3-7.2)^2+(6-7.2)^2+(10-7.2)^2+(3-7.2)^2+(8-7.2)^2+(4-7.2)^2+(8-7.2)^2+(4-7.2)^2+(7-7.2)^2}{15-1}} \\ \\ s=\sqrt{\frac{188.4}{14}} \\ \\ s=\sqrt{13.457} \\ \\ s=3.7 \end{gathered}[/tex]Therefore, the standard deviation is 3.7
xzANSWER:
3.7
what are the x and y intercepts of the linear function given by the equation 2x+5y=-10
In order to find the x-intercept of the function, you need to evaluate the function for y = 0, so:
[tex]\begin{gathered} 2x+5y=-10 \\ y=0 \\ 2x+5(0)=-10 \\ 2x=-10 \\ x=-\frac{10}{2} \\ x=-5 \end{gathered}[/tex]So, the x-intercept is x = -5
In order to find the y-intercept of the function, you need to evaluate the function for x = 0, so:
[tex]\begin{gathered} 2x+5y=-10 \\ x=0 \\ 2(0)+5y=-10 \\ 5y=-10 \\ y=-\frac{10}{5} \\ y=-2 \end{gathered}[/tex]Therefore, the y-intercept is y = -2
Simplify: -12- (–17)
Given the expression:
[tex]-12-(-17)[/tex]You can simplify it as follows:
1. Multiply the signs applying the Sign Rules for Multiplication:
[tex]\begin{gathered} -\cdot-=+ \\ +\cdot-=- \\ +\cdot+=+ \\ -\cdot+=- \end{gathered}[/tex]Then:
[tex]=-12+17[/tex]2. Notice that the signs of the numbers are different. Therefore, you have to subtract them. The result will have the same sign of the number with the greatest absolute value (in this case, it will be positive:
[tex]=5[/tex]Hence, the answer is:
[tex]=5[/tex]R=R¹+R² Solve for R²
Given that ;
R=R¹+R² -------- make R² the subject of the formula as;
Take R¹ to the left side of the equation as;
R-R¹ = R²
So;
Answer:
R² = R - R¹
Step-by-step explanation:
R = R¹ + R²
To make R² as the subject subtract R¹ from both sides.
R - R¹ = R²
need help with a question
We can see than 300 is a constant, so mary has 300 stamps for sure.
Now, in that side of the equation, it is added 2x (variable). So that means she is collecting/adding 2stamps per week (where x is number of weeks).
Now, the other side is 10x, so that must be 10 stamps per week.
No constant means initial stamp is 0.
So, this must be Nick's.
Since both sides are equal and we can solve for x, that means:
when will their collection be equal (in what number of weeks).
Looking at the choices, A seems to be correct.
138°12 CSolve for the area of the sector, to the nearest tenth.14.528.90 173.4452.4
Given Data:
The central angle is, θ = 138.
The radius is, r = 12 cm.
The area of the sector can be calculated as,
[tex]A=\frac{\theta}{360}\times\pi\times r^2[/tex]Substituting the values, the area can be calculated as,
[tex]A=\frac{138}{360}\times3.14\times12^2=173.4[/tex]If one sticker is 10 cents and Lia wants 64 stickers how much money does she have to pay?
Solve this problem using a rule of three
1 stick ------------------------ 10 cents
64 stickers ------------------ x
x = (64 x 10) / 1
x = 640 / 1
x = 640 cents
Lia needs to pay 640 cents
1 dollar ----------------- 100 cents
x ---------------- 640 cents
x = (640 x 1) / 100
x = 640/100
x = 6.4 dollars
Lia needs to pay $6.4 dollars
Hello, what I guess I might want to understand is where to plug in the certain numbers/variables I am given. thank you
Solution
The given equation to get the accumulated amount is
[tex]\begin{gathered} A=Pe^{rt} \\ \text{Where r = rate = 10\%}=\frac{10}{100}=\text{ 0}.1 \\ t\text{ = time in years} \\ P\text{= Amount invested}=\text{ \$6000} \\ A=\text{ Accumulated amount = 2 }\times6000\text{ = \$12000 } \end{gathered}[/tex]Therefore, by substituting in these values, t will be given as
[tex]\begin{gathered} 12000\text{ = 6000}\times e^{0.1\times t} \\ \frac{12000}{6000}=e^{0.1t} \\ 2=e^{0.1t} \\ \ln \text{ 2 = 0.1t} \\ 0.6931471806=0.1t \\ t\text{ =}\frac{0.6931471806}{0.1} \\ t\text{ = }6.931471806 \\ t\approx6.9\text{ years to 1 decimal place} \end{gathered}[/tex]t approximately = 6.9 years to 1 decimal place.
THIS IS DUE TODAY PLEASE HELP ASAP AND PROVIDE AN EXPLANATION ILL GIVE YA 80 PONTS!!! Olivia has read 40 pages of a 70 page book, 60 pages of an 85 page book and 43 of a 65 page book. What is the percentage of pages Olivia has not read? PLEASE PROVIDE AN EXPLANATION
Answer: I believe 65%
Step-by-step explanation:
Add all the pages together from the books. Add all the pages they read. Lastly divide the pages they read by the pages there is in total. You get a decimal. move the decimal two numbers over.
consider functions h and k h(x) = 5x^2-1k(x) = square root 5x+1
Given:
[tex]h(x)=5x^2-1\text{ and }k(x)=\sqrt{5x+1}[/tex]Required:
We need to find the function h(k(x)) and k(h(x)).
Explanation:
[tex]Substitute\text{ }h(x)=5x^2-1\text{ in }k(h(x))\text{ to find }k(h(x)).[/tex][tex]k\lparen h(x))=k(5x^2-1)[/tex][tex]Repalce\text{ }x=5x^2-1\text{ in }k(x)=\sqrt{5x+1}\text{ and substitute in }k\lparen h(x))=k(5x^2-1).[/tex][tex]k\lparen h(x))=\sqrt{5\left(5x^2-1\right)+1}[/tex][tex]=\sqrt{5\times5x^2-5\times1+1}[/tex][tex]=\sqrt{25x^2-5+1}[/tex][tex]=\sqrt{25x^2-4}[/tex][tex]=\sqrt{5^2x^2-2^2}[/tex][tex]k(h(x))=\sqrt{(5x)^2-2^2}[/tex][tex]Substitute\text{ }k(x)=\sqrt{5x+1}\text{ in }h(k(x))\text{ to find }h(k(x)).[/tex][tex]h(k(x))=h(\sqrt{5x+1})[/tex][tex]Repalce\text{ }x=\sqrt{5x+1}\text{ in }k(x)=5x^2-1\text{ and substitute in h}\lparen k(x))=h(\sqrt{5x+1}).[/tex][tex]h(k(x))=5(\sqrt{5x+1})^2-1[/tex][tex]h(k(x))=5(5x+1)-1[/tex][tex]h(k(x))=5\times5x+5\times1-1[/tex][tex]h(k(x))=25x+5-1[/tex][tex]h(k(x))=25x+4[/tex][tex]h(k(x))=5^2x+2^2[/tex]We get
[tex]k(h(x))=\sqrt{(5x)^2-2^2}[/tex]and
[tex]h(k(x))=5^2x+2^2[/tex]We know that
[tex]\sqrt{(5x)^2-2^2}\ne5^2x+2^2[/tex][tex]k(h(x))\ne h(k(x))[/tex][tex]Recall\text{ that if }k(h(x))=h(k(x))\text{ then h and k are inverse functions.}[/tex]Final answer:
[tex]For\text{ x}\ge0,\text{ the value of h\lparen k\lparen x\rparen\rparen is not equal to the value of k\lparen h\lparen x\rparen\rparen.}[/tex][tex]For\text{ x}\ge0,\text{ functions h and k are not inverse functions,}[/tex]