The function represented by the table is: [tex]Y = (1/2)^X[/tex].This is an example of an exponential function.
What is Algebraic function ?
An algebraic function is a function created by applying the operation of addition, subtraction, multiplication, division, and extracting the nth root
To determine the type of function represented by the table, we can look at the pattern of the values of X and Y.
As we can see from the table, as X increases by 1, Y is divided by 2. This means that the relationship between X and Y is not linear but rather exponential. Specifically, the function is of the form Y = [tex]a(1/2)^X,[/tex] where a is a constant.
To find the value of a, we can use one of the points in the table. Let's use the first point, (-4, 16):
16 = [tex]a(1/2)^(-4)[/tex]
Multiplying both sides by[tex](1/2)^(-4)[/tex]:
[tex]16 * (1/2)^4[/tex] = a
a = 16 * (1/16) = 1
Therefore, the function represented by the table is:
[tex]Y = (1/2)^X[/tex]
This is an example of an exponential function.
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HELP PLEASE EASY 20 POINTS!!
After running a marathon,Alvin reduced his training by 18 miles per week on a period of 3 weeks. Which equation can be used to represent the average weekly change in his training?
A) -18 ÷ -3 = +6
B) +18 ÷ -3= -6
C) +18 ÷ +3 = +6
D) -18 ÷ 3 = -6
Micheal is a swimmer. In 2009,he swam the men's 50-meter freestyle in 23.04 seconds. In the same year,he swam the 100 meter freestyle in 47.77 seconds. How much faster,in meters,was his 50-meter freestyle time then his 100-meter freestyle time?
The answer is C) +18 ÷ +3 = +6. Micheal's 50-meter freestyle time was 24.73 meters faster than his 100-meter freestyle time.
What is speed?Measure of how far an object travels in a certain amount of time. Speed is calculated by dividing the distance traveled by the time it took to travel that distance.
The answer is C) +18 ÷ +3 = +6.
This is because Alvin decreased his training by 18 miles over a period of 3 weeks.
Dividing the decrease by the number of weeks gives us 6 miles, which is the average weekly change in his training.
That is 18/3= 6.
To answer the second question, Micheal's 50-meter freestyle was 24.73 meters faster than his 100-meter freestyle.
To calculate this, we subtract the time of the 50-meter freestyle (23.04 seconds) from the time of the 100-meter freestyle (47.77 seconds) to get 24.73 seconds.
We then multiply this time by the speed of a meter per second (1 meter/second) to get 24.73 meters.
Therefore, Micheal's 50-meter freestyle time was 24.73 meters faster than his 100-meter freestyle time.
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Evaluate the expression. (4−7)4 − 52
Answer: -64
Step-by-step explanation:
Choose which function is represented by the graph.
D
I graphed each one and then compared the graphs to the one in the question
5. Which of the following transformations maps Figure A onto Figure B?
A) Translate Figure A 3 units right, and then reflect it across the x-axis.
B) Reflect Figure A across the x-axis, and then translate it 3 units left.
C) Reflect Figure A across the y-axis, and then translate it 3 units right.
D) Translate Figure A 3 units right and 2 units down.
-7654321
2
Figure A
123456
Figure B
Answer:
A)
Step-by-step explanation:
just look at the picture.
what must have happened to figure A to turn into figure B ?
is B more to the right or to the left of A ?
well, right. isn't it obvious ?
and when you look at the coordinates of a vertex, it is 3 units to the right of A.
and if the shift to the right had not happened - the 2 figures would be a mirrored image of each other with the x-axis being the mirror.
that is why A) is the right answer.
Suppose a new cross section was created in each solid, both at the same height, using some scale factor k. How would the areas of these 2 cross sections compare? Explain your reasoning.
The areas of the two cross-sections would be proportional to k².
If a new cross-section was created in each solid, both at the same height using a scale factor k, the areas of the two cross sections would be proportional to k².
The reason for this is that when a scale factor is applied to a two-dimensional figure, the area of the resulting figure increases by a factor of k². This is because the linear dimensions of the figure (length and width) are multiplied by k, and since the area is calculated by multiplying length and width, the area of the figure is multiplied by k².
Therefore, if the two cross sections are created at the same height and are scaled by the same factor k, their areas will be proportional to k². For example, if k=2, the area of each cross-section will be four times the original area; if k=3, the area of each cross-section will be nine times the original area, and so on.
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PLEASE HELP I NEED IT ASAP
Explain using the change of base formula and evaluating the change of base formula
[tex]log_2(100)[/tex] is apprοximately 6.64 when evaluated using the change οf base fοrmula with base 10.
What is the change οf base fοrmula?The change οf base fοrmula is used in lοgarithmic functiοns tο change the base οf the lοgarithm frοm οne value tο anοther.
The fοrmula states that fοr any base b, and any pοsitive numbers x and y, the fοllοwing equatiοn hοlds true:
[tex]log_b(x) = log_y(x) / log_y(b)[/tex]
Here,[tex]log_b(x)[/tex] represents the logarithm of x with base b, and [tex]log_y(x)[/tex]represents the logarithm of x with base y.
Tο evaluate the change οf base fοrmula, we need tο substitute the given values οf x, y, and b in the fοrmula and simplify the expressiοn. Fοr example, let's say we want tο find [tex]log_2(100)[/tex] using the change οf base fοrmula with base 10. We can write:
[tex]log_2(100)[/tex] [tex]= log_{10}(100) / log_{10}(2)[/tex]
Here, we know that log_10(100) = 2, as 10^2 = 100, and log_10(2) is approximately 0.301. Therefore, we can simplify the expression as:
[tex]log_2(100)[/tex] = 2 / 0.301
This gives us:
[tex]log_2(100)[/tex] ≈ 6.64
Therefore, [tex]log_2(100)[/tex] is approximately 6.64 when evaluated using the change of base formula with base 10.
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Complete Question:
How do you use the Change of Base Formula and a calculator to evaluate the logarithm log₂1 ?
Botanists conduct field research to develop recovery plans for endangered species.
They use linear and exponential models and graphs to interpret the resulting data.
Seth tracks the average number of a plant species found in a forest area.
He uses volunteers to help search for the species.
The number n of plants found increases by r percent for each volunteer added to the search. Seth found p plants
before adding any volunteers.
Write an exponential growth function that relates the number n of plants found to the number X of volunteers Seth has.
Exponential growth function that relates the number n of plants found to the number x of volunteers Seth has,
[tex]n = p\left(1+\frac{r}{100}\right)^x[/tex]
How to find an exponential function?
Exponential function is the function of growth or decay with the power of the independent variable.
As, n is the number of plants increased.
r is the percent increase
p is the initial plants
and x is the volunteers
Compound Interest/ Exponential equation is,
[tex]n = p\left(1+\frac{r}{100}\right)^x[/tex]
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The exponential growth function [tex]n = p(1 + r)^X[/tex]
How to express exponential growth function?The exponential growth function that relates the number n of plants found to the number X of volunteers Seth has can be expressed as:
[tex]n = p(1 + r)^X[/tex]
where:
n is the number of plants found
p is the initial number of plants found before adding any volunteers
X is the number of volunteers added to the search
r is the percentage increase in the number of plants found per volunteer
he growth factor is represented by the term (1 + r), which is multiplied by the initial number of plants discovered to determine the new number of plants discovered when each volunteer is added. The value of (1 + r)X increases exponentially as X increases, resulting in an exponential increase in the number of plants found.
It is essential to keep in mind that the growth rate r in the formula must be expressed as a decimal rather than a percentage. For instance, r = 0.05 if the number of plants discovered increases by 5% for each volunteer added.
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claims filed under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000. what is the probability that the average of 25 randomly selected claims exceeds 20,000?
To find the probability that the average of 25 randomly selected claims exceeds 20,000, we need to first determine the mean and standard deviation of the sample distribution.
Given:
- Mean of the population (μ) = 19,400
- Standard deviation of the population (σ) = 5,000
- Sample size (n) = 25
Step 1: Calculate the mean of the sample distribution (μ_sample)
Since the sample mean is an unbiased estimator of the population mean, μ_sample = μ = 19,400.
Step 2: Calculate the standard deviation of the sample distribution (σ_sample)
σ_sample = σ / √n = 5,000 / √25 = 5,000 / 5 = 1,000
Now, we need to find the probability that the sample mean exceeds 20,000. We can do this using the Z-score formula:
Z = (X - μ_sample) / σ_sample
where X is the sample mean we want to find the probability for (20,000 in this case).
Step 3: Calculate the Z-score
Z = (20,000 - 19,400) / 1,000 = 600 / 1,000 = 0.6
Step 4: Find the probability
Now, we need to find the area to the right of the Z-score in the standard normal distribution table or use a calculator/software that provides the probability.
The area to the right of Z = 0.6 is approximately 0.2743. So, the probability that the average of 25 randomly selected claims exceeds 20,000 is approximately 0.2743 or 27.43%.
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O is the center of the regular decagon below, the radii equals 16. Find its perimeter. Round to the nearest tenth if necessary.
The length of the side is 67.8 which gives us the perimeter as 67.8 x 10 = 678units
We are given a decagon, which is a 10-sided polygon with center O, and a radius of 11 units.
We are required to find the perimeter of the decagon and round the answer to the nearest tenth.
We begin by calculating the interior angle of e decagon.
We use the formula;
S = (n-2) 180, where,
S = sum of interior angles,
n = number of sides
therefore, S = (10 - 2) 180
S = 8 x 180
S = 1440
For a 10-sided polygon, each interior angle will now measure;
Interior angle = 1440/10 = 144
take note that the radius of the polygon divides the angle at the vertex (that is, the interior angle) into two equal halves.
With that in mind we can now construct the following triangle:
Take note that the line from the center to the side is the apothem, while the side labeled 11 is the radius (which was given). We can now extract one of the right angled triangles and use that calculate the apothem as follows:
The side labeled a; the apothem can be calculated using the angle 72 degrees as the reference angle:
reference angle = 72
opposite = a
hypotenuse = 11
therefore when we use the formula of sin, we get a = 10.46
Also, for the side labeled s (half the length of one side of the decagon); s = 3.39
For the length of one side of the decagon, we have the side s times 2. Hence, the length of the side is 67.8
67.8 x 10 = 678units
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the ratio of centimeters to meters is 100:1. laura has a rope that is 40 cm. how long is the rope in meters?
Answer:
0.4 meters.
Step-by-step explanation:
The ratio of centimeters to meters is 100:1.
That means if there are 100 centimeters, there is 1 meter.
So:
centimeters divided by 100 is a meter.
In this case, we have 40 centimeters.
Since there are less than 100 centimeters, we can't divide by 100, so we go into the decimals.
40 divided by 100 is
40/100
simplified to:
4/10
turned into decimal:
0.4 meters.
Hope this helps :)
The answer is 0.4 meters
The given ratio of centimeters to meters is 100:1. The rope Laura has is 40 cm long. To determine how long the rope is in meters, we need to divide 40 cm by 100 to convert it to meters.
Therefore, the length of the rope in meters is 0.4 meters (40/100 = 0.4).Answer: 0.4 meters
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PLEASE HELP THIS IS URGENT
Answer:
7 hours and 2 hours longer
The volumes of both containers are 6,912 cubic inches.
What is the height of container 2?
Set up an equation using the width and height from con-
tainer 2 to find the missing height.
V = lwh
Container 1
32 in.
12 in.
18 in.
Container 2
24 in.
? in.
16 in.
Answer:
let the height of second container be x
Then volume=6912
we know that volume of cuboid= lbh
so ，
24.x.16=6912
384.x=6912
x =6912divide 384
therefore x =18 that is the height of the container
Let ²-13x - 30 = (x+p) (x+q). If the absolute value of p is greater than the absolute
value of q, which number, p or q, is a negative integer? Explain your answer.
Answer: We can use the fact that the sum and product of the roots of a quadratic equation are related to its coefficients, and use this relationship to find the values of p and q.
Given the equation ²-13x - 30 = (x+p)(x+q), we can see that the coefficient of the x^2 term is 1, the coefficient of the x term is -13, and the constant term is -30.
By comparing the coefficients with the formula for the sum and product of the roots of a quadratic equation, we have:
p + q = -(-13)/1 = 13
p*q = -30/1 = -30
Since the absolute value of p is greater than the absolute value of q, we know that either p is positive and q is negative, or p and q are both negative. We can eliminate the possibility of p being positive and q being negative, since their product would be negative, and we know that p*q = -30. Therefore, both p and q must be negative integers.
We also know that the sum of p and q is 13, which means that the absolute value of q is less than the absolute value of p. Since p and q are both negative, this means that q has a larger absolute value (i.e., is farther from zero) than p. Therefore, q is the negative integer in this case.
Step-by-step explanation:
The heights of 11 plants, in inches, are listed. 14, 15, 16, 16, 17, 17, 17, 18, 18, 19, 22 If another plant with a height of 14 inches is added to the data, how would the range be impacted? The range would decrease to 8 inches. The range would stay the same value of 8 inches. The range would increase to 17 inches. The range would stay the same value of 18 inches.
The range wοuld stay the same value οf 8 inches.
What is average?Let's lοοk at the average fοrmula in mοre detail in this part and use sοme examples tο illustrate hοw it may be used. The fοllοwing is an example οf the average fοrmula fοr a specific set οf data οr οbservatiοns: Average = (Sum οf Observatiοns) ÷ (Tοtal Numbers οf Observatiοns).
In the οriginal data set, the maximum value is 22 inches and the minimum value is 14 inches, sο the range is:
22 - 14 = 8 inches
If anοther plant with a height οf 14 inches is added tο the data, the minimum value will still be 14 inches, but the maximum value will nοw be 22 inches (since there are twο plants with a height οf 22 inches).
Therefοre, the range will increase:
22 - 14 = 8 inches
Sο the cοrrect answer is: The range wοuld stay the same value οf 8 inches.
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For each of the figures write an absolute value equation that has the following solution set.
The absolute value equation | x - (-5) | = 3 has a solution set corresponding to a straight horizontal line with marked points -8, -2, and x in left to right manner.
How to write an absolute value equation that has the following solution set?
The solution set given as a straight horizontal line passing through the points -1/2 and 3/2 can be represented as:
| x - 1 | = 1/2
Explanation:
The absolute value function | x - 1 | gives the distance between x and 1 on the number line.Since the given solution set is a straight horizontal line passing through -1/2 and 3/2, the midpoint of the two points would be 1.The distance between the midpoint (1) and one of the points (-1/2 or 3/2) would be 1/2.Therefore, the equation | x - 1 | = 1/2 represents the set of all values of x that are at a distance of 1/2 units away from the midpoint 1, which is the straight horizontal line passing through -1/2 and 3/2.To learn more about horizontal line, visit: brainly.com/question/30197744
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What is the value of x? Enter your answer in the box.
Thus, the value of the variable x for the given one variable linear equation is found as: x = 8.5.
Explain about the one variable linear equation:A linear equation is a one-variable equation of such a straight line. The variable only has one power, which is 1. Simple algebraic operations are used to solve linear equations in one variable, which can have the form ax+b=0.
All values which it make this equation true are included in the solution set. All real numbers make up the solution set for this equation because, when a real number is used in place of x, the equation is proved to be true.
Given one variable linear equation :
3x + 8 = 5x - 9
Isolate the variable and the constants on the different sides;
5x - 3x = 8 + 9
2x = 17
x = 17/2
x = 8.5
Thus, the value of the variable x for the given one variable linear equation is found as: x = 8.5.
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Complete question:
What is the value of x? The given equation is: 3x + 8 = 5x - 9.
Write a proportion comparing the rise to the run for each of the similar slope triangles and find the numeric value.
The proportion of the rise to the run for each of the similar slope triangles is given by:
h / b = mh' / mb' = mh / b
What is proportion?
A proportion is a statement that two ratios or fractions are equal. It is commonly written in the form of two fractions separated by an equal sign, such as a/b = c/d.
To write a proportion comparing the rise to the run for each of the similar slope triangles, we can use the fact that the ratio of corresponding sides of similar triangles is the same.
Let's say we have two similar triangles with corresponding sides of length a, b, and c, and a', b', and c', respectively. Then we can write the following proportion:
a / a' = b / b' = c / c'
Now, let's apply this to finding the proportion of the rise to the run for each of the similar slope triangles.
In a right triangle, the slope is defined as the ratio of the rise (vertical change) to the run (horizontal change). Let's say we have two similar right triangles with slopes m and m', respectively, and the rise and run of the first triangle are h and b, respectively. The rise and run of the second triangle are then mh and mb', respectively.
We can write the proportion of the rise to the run for each triangle as:
h / b = mh' / mb'
Simplifying this proportion, we can cancel out the common factor of b:
h / b = mh' / mb'
h / 1 = mh' / m'
h = bmh'
Therefore, the proportion of the rise to the run for each of the similar slope triangles is given by:
h / b = mh' / mb' = mh / b
The numeric value of this proportion will depend on the specific values of the rise and run for each triangle and the slope of the triangles.
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c/0.5 - 3.2 = 2.6 I dont get it
Answer:
c/0.5= 2.6+3.2
c= 5.8×0.5
c=2.9
what’s the answer???
Answers and explanation:
[tex]3x^{2} +12x+9=(3x+3)(x+3)[/tex]
Factored form:
[tex](3x+3)(x+3)=0[/tex]
First zero:
[tex]3x+3=0[/tex]
[tex]3x=-3[/tex]
[tex]x=\frac{-3}{3} =-1[/tex]
Second zero:
[tex]x+3=0[/tex]
[tex]x=-3[/tex]
Ordered pair:
[tex]x=-1, x= -3[/tex]
Hope this helps.
The perimeter of a rectangle is 88, and the length is 1 less
than 2 times the width. What is the length?
O 29
O24
O 32
O 15
Answer:
29
Step-by-step explanation:
Given:
P = 88
Let's assume, that the length is x and the width is y
x = (2y - 1)
Let's write an equation for the perimeter:
2x + 2y = 88
2 × (2y - 1) + 2y = 88
4y - 2 + 2y = 88
6y = 88 + 2
6y = 90 / : 6
y = 15
We found the width, now we can find the length:
x = 2 × 15 - 1 = 30 - 1 = 29
Given h(x) = 4x + 3, find h(5)
*60 POINTS FOR FOUR CORRECT GEOMETRY ANSWERS*
Please answer these for me. Thank you
The length of the altitude is
B. 7.5The similarity statement about the triangles is
c. triangle MON ~ triangle MPO ~ triangle OPNThe value of x in the pentagon is 21
How to find the value of xThe value of x is solved by applying the the proportions of the pentagon
20 / 15 = 18 / 13.5 = 4/3
therefore, 12 * 4/3 = x - 5
16 = x - 5
16 + 5 = x
x = 21
The altitude of the right triangle
h / 14 = 4 / h
h^2 = 14 * 4
h^2 = 56
h = sqrt (56)
h = 7.5
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Can someone solve this for me please I need it fast
The value of the indicated sides of the shape given above are as follows:
Scale factor = 1.67
X = 15
Y = 9
Z = 9
How to calculate the scale factor of the given shape above?To call the scale factor of a given shape the formula used is given below;
Scale factor = Dimension of new shape(bigger)/dimension of the old shape(smaller).
Dimension of new shape = BC = 30
Dimension of old shape = FE = 18
scale factor = 30/18 = 1.67
X = AD/1.67
X = 25/1.67
X = 15 (approximately)
Y = DC/1.67
Y = 15/1.67
Y = 9 (approximately)
Z = AB/1.67
Z = 15/1.67
Z = 9(approximately)
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please help need both answers please will give 30 points please i need both answered i dont need work shown i just need both answers please
Answer:
Angle A = Angle L
Angle D = Angle M
Angle C = Angle N
Segment AB = Segment LB
Segment CD = Segment NM
Segment DA = Segment ML
2nd part
Quadrilateral URST has moved by a translation of (x-5,y+2)
Step-by-step explanation:
Answer:
Step-by-step explanation:
Write the equation of a line that contains the given point and has the given slope:
(-8,-8), slope is 3
Answer:
Ok
Step-by-step explanation:
Y = mx + c
Slope is for the gradient =m therefore we have +3 as our gradient
Now we are going to substitute pont - 8mac- 8 on the above equation
Y = mx +c
-8=3(-8) +c
-8=-24 +c
-8+24 =c
16=c
This means that the y intecept of this straight line is 0 mac 16
Meaning that when the line touches the y axis at 16 x will be equal to zero
Therefore our equation is
Y =3x +16
A bag contains 8 red marbles, 3 blue marbles, and 4 green marbles. What is the probability
. Carlos draws a green marble, does not replace it, and then draws another green marble?
4/15
16/225
54/210
2/35
To solve this problem, we use the rule of conditional probability which states that the probability of the joint event A and B happening is equal to the probability of A happening multiplied by the probability of B happening given that A has already happened.
So, the probability of drawing a green marble on the first draw is 4/15, since there are 4 green marbles out of a total of 15 marbles.
If the first marble drawn is green and not replaced, there are now 14 marbles remaining in the bag, out of which 3 are green.
Thus, the probability of drawing another green marble given that the first one was green is 3/14.
Therefore, the probability of drawing two green marbles without replacement is:
(4/15) * (3/14) = 2/35
So the answer is 2/35.
Create a real life example that includes function composition in it and solve it?
how can you multiply binomial with a trinomial? Give an example about it and solve it.
For instance: Let f(x) = 2x + 1 and g(x) = x^2. Find (f ∘ g)(3). First, we need to find g(3) which is 3^2 = 9. Then we plug this value into f: f(g(3)) = f(9) = 2(9) + 1 = 19. Therefore, (f ∘ g)(3) = 19.
What is the description of a function composition?Function composition is a mathematical concept that is used to combine two or more functions into a single function. This concept is widely used in real life situations, such as in computer programming, engineering, and physics.
For example, in computer programming, function composition is used to break down complex programs into smaller, more manageable parts. By using function composition, programmers can create complex programs that are easier to read and maintain. Therefore, function composition is a powerful tool that is used in various fields to simplify complex problems and create more accurate models of the world around us.
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jane is planning to offer a groupon for inner tube rentals that she will distribute on hot, sunny, summer days by the river that runs through her town. based on her past experience with groupon, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day. if jane would like her expected revenue to be at least $390 per day, what should the groupon price be? (round your answer up to the nearest whole dollar amount.)
The price of Groupon for revenue of $300 is $4 if the expected revenue is at least $390 per day.
Probability is assigned as
x =30, 60, 120, 180
P(x) = 0.10,0.40, 0.40, 0.10
Expected sales volume:
Number of tubes x =30, 60, 120, 180
Probability P(x)= 0.10,0.40, 0.40, 0.10
Expected values are 3, 24,48,18
Total = 93 tubes
Groupon price = $390/$93 = $4.19
Jane's price for each Groupon is the daily rental income divided by the expected number of tubes rented per day. The expected number of tubes is derived by multiplying the expected number of each tube by its probability and then summing the results.
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The diagram shows parallel lines cut by two transversal lines creating a triangle. Which statement can be made from the diagram?
mAngleB + mAngleC + 55° = 180°
mAngleA + 60° + 55° = 180°
mAngleC = 55°
mAngleA = 55°
Answer:
The answer to your problem is, B. mAngleA + 60° + 55° = 180°
Step-by-step explanation:
Let the Triangle be ΔABC , such that
∠A + ∠B + ∠C = 180°
The area of the triangle = ( 1/2 ) x Length x Base
For a right angle triangle
if a² + b² = c² , it is a right triangle
if a² + b² < c² , it is an obtuse triangle
if a² + b² > c² , it is an acute triangle
Let the triangle be represented as ABC:
Now , the first measure of angle = 55
The second measure of angle = 60
The angle of the first ∠A = 180
Substituting the values in the equation , we getFor a triangle , ∠A + ∠B + ∠C = 180°So , sum of the angles of a triangle mAngleA + 60° + 55° = 180°Thus the answer to your problem is, B. mAngleA + 60° + 55° = 180°
Math help
Nyssa made the table below to show the relation between the side length and the areas of various poster boards
The relation between the area (a) and the side length (s) in the table is:
A. a = s²
What is an area?
To see this relationship, we can observe that the areas of the poster boards are the square of their respective side lengths. For example, when the side length is 2 feet, the area is 4 square feet (2²).
Similarly, when the side length is 4 feet, the area is 16 square feet (4²). This pattern holds true for all the side lengths listed in the table. Therefore, we can conclude that the area of each poster board is equal to the square of its side length.
Calculation:
Side Length (s) | Area (a)
1 | 1
2 | 4
3 | 9
4 | 16
As we can see, the area of each poster board is equal to the square of its side length. Thus, the relation between the area (a) and the side length (s) in the table is a = s².
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Answer:
A
Step-by-step explanation:
[tex]a = s {}^{2} \\ 1. \: + - \sqrt{a = s} \\ 2. \: s = + - \sqrt{a} [/tex]