Answers
There are a couple ways to go about this.
By definition of inverse function, if [tex]j^{-1}[/tex] exists, then
[tex]j\left(j^{-1}(x)\right) = x[/tex]
All this means is that if you provide some input [tex]x[/tex], plug it into [tex]j^{-1}(x)[/tex], then plug that into [tex]j(x)[/tex], you end up recovering the original input [tex]x[/tex] that you started with.
We can also do this in the opposite direction, so
[tex]j^{-1}(j(x)) = x[/tex]
So now if [tex]j(x)[/tex] is invertible, then
[tex]j(x) = -\dfrac23 x - 9 \implies j\left(j^{-1}(x)\right) = -\dfrac23 j^{-1}(x) - 9 = x[/tex]
Solve for [tex]j^{-1}[/tex].
[tex]-\dfrac23 j^{-1}(x) - 9 = x[/tex]
[tex]\dfrac23 j^{-1}(x) = -x - 9[/tex]
[tex]j^{-1}(x) = -\dfrac32 (x+9)[/tex]
[tex]\boxed{j^{-1}(x) = -\dfrac32 x - \dfrac{27}2}[/tex]
Another way to do it is by algebraically substituting [tex]x[/tex] within the definition of [tex]j(x)[/tex]
Starting with the given definition
[tex]j(x) = -\dfrac23 x - 9[/tex]
we replace [tex]x[/tex] with some new expression in a new variable [tex]y[/tex] that makes the right side reduce to just [tex]y[/tex]. To demonstrate, if we multiply [tex]-\frac23 x[/tex] by [tex]-\frac32[/tex], we end up with a coefficient of 1. So let [tex]x=-\frac32 y[/tex]. Then
[tex]j\left(-\dfrac32 y\right) = -\dfrac23 \left(-\dfrac32 y\right) - 9 = y - 9[/tex]
As you can see, this doesn't completely reduce to [tex]y[/tex], so we're not done. To adjust for this, we just add 9 to [tex]y[/tex], so our ultimate substitution is [tex]x=-\frac32(y+9)[/tex]. Then
[tex]j\left(-\dfrac32 (y+9)\right) = -\dfrac23 \left(-\dfrac32 (y+9)\right) - 9 = (y+9) - 9 = y[/tex]
Now applying [tex]j^{-1}[/tex] to both sides,
[tex]j^{-1}\left(j\left(-\dfrac32 (y+9)\right)\right) = j^{-1}(y)[/tex]
[tex]\implies \boxed{j^{-1}(y) = -\dfrac32 y - \dfrac{27}2}[/tex]
(we can freely swap out [tex]y[/tex] for [tex]x[/tex] at this point)
Yet another way, since the function here is quite simple, we can just think about the action of [tex]j[/tex] on a given number [tex]x[/tex] - multiply it by -2/3, and subtract 9.
[tex]x \mapsto -\dfrac23 x \mapsto -\dfrac23x - 9 = j(x)[/tex]
The inverse operation undoes this, so given some number [tex]j(x)[/tex], we recover [tex]x[/tex] by adding the additive inverse of -9, or 9, then multiplying the whole result by the multiplicative inverse of -2/3, or -3/2.
[tex]j(x) \mapsto j(x) + 9 \mapsto -\dfrac32(j(x) + 9) \mapsto -\dfrac32 j(x) - \dfrac{27}2 = x[/tex]
You seem to be using some version of this method. You made the mistake of thinking the "multiply by -3/2" step applies only to [tex]j(x)[/tex], and not [tex]j(x) + 9[/tex].
Related Questions
8. 8x-2y 10 xy (if x = 4 and y = -7)
Answers
You are backpacking through Europe. You have 30 days until your flight home, but you have spent all
have found a hotel that is willing to accept one link per night for payment of room and board. However,
the manager
that they want one gold link in payment for every link they cut. How many links will you have when
you call home?
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Answer:
29
Step-by-step explanation:
Itvis the correct answer in my opinion
Determine the truth value of the following conditional statement. If true, explain your reasoning. If false , give a counterexample.
If pigs can fly, then 2+5=7 .
Answers
The truth value of the statement; If pigs can fly, then 2+5=7 is false.
What is syllogism?The term syllogism has to do with logical reasoning. In this case, the premise must lead to the conclusion. The premise is the basis upon which we could draw the conclusion. The truth value of the premise is what determines the truth value of the conclusion as one naturally follows from the other.
Now, in this case we have a conditional statement. Recall that a conditional statement is an if – then statement. The truth value of the premise has an impact on the truth value of the conclusion. We have the statement written in the manner; If pigs can fly, then 2+5=7 .
We know that pigs can not fly. This is a false premise. However, the conclusion that has been given is true by the principle of addition in mathematics. The truth value of the statement is false.
An example of a truth value in which both the premise and the conclusion are true is; If lions are wild animals then lions are one of the animals found in the forest.
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one (friend a) talks about computer science 80% of the time, and linguistics 20% of the time; the other (friend b) talks about linguistics 70% of the time, and computer science 30% of the time. one day, you find a typed note on your desk about computer science. your lab-mates leave you notes equally often, so you don't know who left this one. what is the probability the note is from friend a? show your work.
Answers
40% is the probability the note is from friend A.
The lab mates leave notes equally often,
Probability of A 50%
Probability of B 50%
Probability A talks about computer science at 80%
Probability B talks about computer science at 30%
chance that the note is from a friend A is
A = 50% X 80%
= 40%
Hence, 40% is the probability the note is from friend A
What do you mean by probability?
Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about how probable an event is to happen, or its chance of happening.
By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula.
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-2-12(-2x-3) apply distributive property
Please i need asap!!
Answers
I think this is the answer, I totally forgot what distributive property is.
wait. im not so sure, ignore me if you don't think im right
need help solving please
Answers
Answer:
b
Step-by-step explanation:
15^2= 225
[tex]5\sqrt{3}= 8.66025404\\8.66025404^{2} = 75\\225+75=300\\\\10\sqrt{3}=17.3205081\\17.3205081{2}= 300\\\\225+75=300\\300=300[/tex]
There are 32 students in the middle school choir. How can this number be written as a power with a base of 2? Explain how you found your answer.
Answers
The number 32 can be written as a power with a base of 2 as 2^5
What is index form?The index form of a number is defined as that number represented as an exponential expression, or even a single number raised to another number.
Example;
2^n
3^3
From the information given, we have to determine the index form of 32
It is expressed in the base of 2 as;
32 = 2 × 2 × 2 × 2 × 2
32 = 2^5
Thus, the number 32 can be written as a power with a base of 2 as 2^5
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Describe the sample variance using words rather than a formula. do the same with the population variance.
Answers
Sample variance is a term that be defined as the outcome of the squared difference of any given data points that is said to be obtained from the mean of the data set.
What is the sample variance?It is also known to be a form of an absolute measure of a dispersion and is known also to be used to examine the deviation of data points with regards to the data's average.
Population variance is a term that is defined as a measure of the de spread of population data.
Therefore, Sample variance is a term that be defined as the outcome of the squared difference of any given data points that is said to be obtained from the mean of the data set.
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Find the measure of angle b
66°
b
Answers
The measure of angle is 24°.
We are given the two sides of the triangle as:
25 units and 10 units
Since the able that need to be found lies at 25 units side, the base side will be that only.
So, we get that:
Base side = B = 25 units
Perpendicular side = P = 10 units
We know that:
tan θ = P / B
tan θ = 10 / 25
tan θ = 2 / 5
θ = tan⁻¹ ( 2/ 5)
which will be approximately equal to
θ = 24°
Therefore, we get that, the measure of angle is 24°.
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Your question was incomplete. Please refer the content below:
Find the measure of the indicated angle to the nearest degree
a. 66
b. 22
c. 24
d. 68
Translate the sentence into an inequality. twice x is at most 25. what is the answer
Answers
The answer is [tex]2x\leq 25[/tex] .
An inequality compares two values and indicates whether one is less than, greater than, or equal to another.a ≠ b indicates that a is not equal to ba < b indicates that a is less than ba > b indicates that a is greater than b (These two are known as strict inequalities)a ≤ b means that a is less than or equal to ba ≥ b means that a is greater than or equal to b.It is given that twice of x is at most .
“At most” refers to a maximum amount.
“At most” means that any number less than the number presented is acceptable or true,
In mathematically , it is represented as
[tex]2x\leq 25[/tex]
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Use this to calculate the area of a square that has perimeter 6 units
Answers
Answer: daddy 123 456
Step-by-step explanation:
696969696
If (a − b)² = 40 and ab = 8, find a² + b².
Answers
Answer:
a² + b² = 56
Step-by-step explanation:
(a - b)² = a² - 2ab + b² ← substitute values
40 = a² - 2(8) + b²
40 = a² - 16 + b² ( add 16 to both sides )
56 = a² + b²
Looking Ahead Vocabulary
28. The current of a river flows north at a constant rate. What is the constant in the
mathematical expression 3x + 5y +3?
Answers
Answer:
The constant would be 3 in the question
Step-by-step explanation:
question what is an expression equivalent to 6(24) using the distributive property? drag and drop the appropriate number into each box.
Answers
6(20+4) is the equivalent to 6(24) using the distributive property.
How to find an expression equivalent to 6(24) using the distributive property?The distributive Property states that it is necessary to multiply each of the two numbers by the factor before performing the addition operation when a factor is multiplied by the sum or addition of two terms. This property states that multiplying the total of two or more addends by a number will produce the same outcome as multiplying each addend by the number separately and then adding the results together.
In other words, an expression of the type
A (B + C) can be resolved as A (B + C) = AB + AC in accordance with the distributive property.
This characteristic also holds for subtraction.
A(B-C) = AB-AC
given that 6(24)
now compare the 6(24) with general form of distribution property a(b+c) = ab + ac
a = 6 and b+c = 24
Consequently, we must identify two integers the sum is 24.
These numbers 20 and 4 are suitable.
so,
6(20+4) = 6(20) + 6(4)
144 = 144
Hence, the 6(20+4) is the equivalent to 6(24) using the distributive property.
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Quadrilateral WXYZ is a rectangle.
If X W=3, W Z=4 , and X Z=b , find Y W .
Answers
The length of the diagonal YW of rectangle WXYZ is; 5 units.
How to find the diagonals of a rectangle?
The diagonals of a rectangle are congruent and so;
YW ≅ XZ
Thus; YW = XZ = b
Now, given the three sides of right triangle XWZ, we can solve for XZ using Pythagorean Theorem:
(XW)² + (WZ)² = (XZ)²
Plugging in the relevant values gives;
3² + 4² = b²
9 + 16 = b²
25 = b²
b = √25
b = 5
Thus, we can conclude that the length of the diagonal YW is 5 units.
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What is 2,567,653,987+2,000,000,245
Answers
Answer: 4,567,654,232
Step-by-step explanation:
2,567,653,987+2,000,000,245=
4,567,653,987+245=4,567,654,232
10-5/1 +7n ≥-30
Help me this hard
Answers
Answer:0.289x100
Step-by-step explanation:
Answer:
n ≥ - 5
Step-by-step explanation:
10 - 5/1 + 7n ≥ - 30 , simplifying left side
10 - 5 + 7n ≥ - 30
5 + 7n ≥ - 30 ( subtract 5 from both sides )
7n ≥ - 35 ( divide both sides by 7 )
n ≥ - 5
Can someone help me I’m confused
Answers
9/10 miles in 5 minutes how many miles in 1 minute
Answers
Answer:
in 5 minutes=9/10 miles
In 1 minutes= 9/10/5 miles
= 9/10*1/5
=9/50 miles
Please help me please help me
Answers
Answer: A
Step-by-step explanation: Translate up 1 unit, over to the right 5
the points (0, 5) and (0, −5) are the endpoints of the diameter of a circle. the point (3, �) is on the circle, in quadrant 4. what is the value of y?
Answers
Using the equation of a circle, it is found that the value of y is -4.
Equation of the circle:
A circle is a closed curve that is drawn from the fixed point called the center, in which all the points on the curve are having the same distance from the center point of the center.
The equation of a circle with (h, k) center and r radius is given by:
(x-h)² + (y-k)² = r²
Given,
The points (0, 5) and (0, −5) are the endpoints of the diameter of a circle.
The point (3, y) is on the circle, in quadrant 4.
Here we need to find the value of y.
The center is the midpoint of them, thus:
[tex]x=\frac{0+0}{2}=0\\ y=\frac{5-5}{2}=0[/tex]
So, the midpoint is (0,0).
The diameter(twice the radius) is the distance between these two points, so:
[tex]2r=\sqrt{(0-0)^2+(5-(-5))^2}[/tex]
=> 2r = √0+100
=> 2r = √100
=> 2r = 10
=> r = 5
Thus, the equation of the circle is:
=> x² + y² = r²
Apply the values,
=> x² + y² = 5²
=> x² + y² = 25
The point (3,y) is on the circle, in quadrant 4. This means that:
Replacing x by 3, we can find the value of y.
Quadrant 4 means that y < 0.
Then:
=> 3² + y² = 25
=> y² = 25 - 9
=> y² = 16
=> y = ±√16
=> y = - 4
The value of y is -4.
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can someone help me really quick
Answers
Answer:
last one h babysat for 6 hours
Point Y is the midpoint of XZ . Z is the midpoint of YW. Prove that XY ≅ ZW.
Answers
From equation 1 and 2, you can see that XY = YZ and Z = ZW, hence according to the transitive property, we can conclude that XY ≅ ZW.
Midpoint of coordinateA midpoint is a point that divides a line of a segment into two equal parts. It also means that the point bisects the given line.
If the point Y is the midpoint of XZ, then;
XY + YZ = XZ and;
XY = YZ .........1
Similarly, if Z is the midpoint of YW, then;
YZ + ZW = YW and;
YZ = ZW ..........2
From equation 1 and 2, you can see that XY = YZ and Z = ZW, hence according to the transitive property, we can conclude that XY ≅ ZW.
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Write 7^-6 using a positive exponent.
Answers
Answer:
7^-6
1/7^6 ( using law of negative index)
Answer:
Other person is correct.
Step-by-step explanation:
:>
A 9,000 deposit for 4 years at a compound interest rate of 6%
Answers
Answer:
Your answer is $11,362
I hope this helps!
who wants some points?
Answers
Answer:
4. a plan to win debate about the novels main message, and your favorite words 4. a plan to win debate about the novels main message, and your favorite words 4. a plan to win debate about the novels main message, and your favorite words
Step-by-step explanation:
A car travels 55 miles per hour
1. The unit rate speed for this car is ___ miles per hour
2. if the car has driven 1/2 (0.5) hours, it traveled ___ miles
3. if the car has driven 1 1/2 (1.5) hours, it has traveled ___ miles
4 If the car has traveled 110 miles, it will take ___ hours
!! No explanation needed please! :) correct answer gets brainlesst !!
Answers
For the car travelling with 55 miles per hour-
unit rate = 55 miles per hour.distance travelled = 275 miles.distance travelled = 825 miles.Time taken = 2 hoursWhat is distance speed relation?The distance is a actual length of a path taken by the object.
The speed is the rate at which the distance changes over time. If 'D' is the distance traveled by an object in time 'T,' the speed is equal to s = D/T. It is measured in the same units as velocity.For, the values given in the question;
1. The unit rate speed for this car is ___ miles per hour
S = D/T
S = 55 miles per hour.
Rate of speed is the same as the speed.
Thus,
The unit rate speed for this car is 55 miles per hour
2. if the car has driven 1/2 (0.5) hours, it traveled ___ miles
S = D/T
D = S×T
S = 55 miles per hour.
T = 0.5
Put the values;
D = 55×0.5
D = 275 miles.
Thus, the car travelled 275 miles.
3. if the car has driven 1 1/2 (1.5) hours, it has traveled ___ miles
S = D/T
D = S×T
S = 55 miles per hour.
Put the values;
D = 55×1.5
D = 825 miles.
Thus, the car travelled 825 miles.
4 If the car has travelled 110 miles, it will take ___ hours
S = D/T
S = 55 miles per hour.
D = 110 miles
T = D/S
Put the values;
T = 110/55
T = 2 hours.
Thus, the time taken by car is 2 hours.
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B = {1,
2}ional
(3, 5, 7, 9
Order each set of numbers from least to greatest.
8. √200, 14, 41/3
Answers
The set of numbers ordered from least to greatest is- 41/3, 14, √200.
Here, we are given a set of numbers- √200, 14, 41/3
In order to compare the numbers, we first need to convert all of them into decimal or integer numbers.
Let us analyze them one by one
1. √200
we know that √100 = 10
⇒ √200 = 10√2
√2 = 1.414
Thus, 10√2 = 14.14
2. 14
As we can clearly see 14 < 14.14
⇒ 14 < √200
3. 41/3
Dividing 41 by 3, we get-
13.66
Now, we can see that 13.66 < 14 < 14.14
⇒ 41/3 < 14 < √200
Thus, the set of numbers ordered from least to greatest is- 41/3, 14, √200.
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Fill in the table using this function rule.
y=5x+3
Answers
Answer:
Step-by-step explanation:
5(1) + 3 = 5 + 3 = 8
(1,8)
5(3) + 3 = 8 + 3 = 11
(3, 11)
5(9) + 3 = 45 + 3 = 48
(9, 48)
5(10) + 3 = 50 + 3 = 53
(10, 53)
please help i cant do it rn
Answers
4/5 = 28/35
30/35 - 28/35 = 2/35
Answer:
2/35
Step-by-step explanation:
Given that,
→ (6/7) - (4/5)
Let's solve the problem,
→ (6/7) - (4/5)
→ (30/35) - (28/35)
→ (30 - 28)/35
→ 2/35
Thus, the answer is 2/35.