The interest is a) $7000
b) $17709.73
c) $18672.62
d) $18901.67
What is the formula for simple and compound interest?
Simple interest = (P× r× t)
Compound interest = P(1+r/n)^nt - P
We will find the interest as shown below:
P=$4,000
t=25 years
a) r=7%=0.07
Simple interest = (P× r× t)
= (4000×0.07×25)
= $7000
b) r=7%=0.07
Compound interest = P(1+r)^t - P
= 4000(1+0.07)^25-4000
= $17709.73056
rounding to nearest cents
= $17709.73
c) r=7%=0.07
n=4
Compound interest = P(1+r/n)^nt - P
= 4000(1+0.07/4)^(25*4)-4000
= $18672.62375
rounding to nearest cent
= $18672.62
d) r=7%=0.07
n=12
Compound interest = P(1+r/n)^nt - P
= 4000(1+0.07/12)^(25*12)-4000
= $18901.6728
rounding to nearest cent
= $18901.67
Hence, the interest is a) $7000
b) $17709.73
c) $18672.62
d) $18901.67
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There are 81 athletes who joined a parade. Four-ninths of them are basketball players. How many basketball players joined the parade?
Answer:
36
Step-by-step explanation:
81 x (4/9) =
36
in how many ways can the letters in the word payment be arranged if the letters are taken 6 at a time
Considering the definition of permutation, in 5040 ways can the letters in the word ''PAYMENT'' be arranged if the letters are taken 6 at a time.
What is PermutationPermutation is placing elements in different positions. So, permutations of m elements in n positions are called the different ways in which the m elements can be arranged occupying only the n positions.
That is, permutations refer to the action of arranging all the members of a set in some sort of order or sequence.
In other words, permutations (or Permutations without repetition) are ways of grouping elements of a set in which:
take all the elements of a set.the elements of the set are not repeated.order matters.To obtain the total of ways in which m elements can be placed in n positions, the following expression is used:
[tex]mPn=\frac{m!}{(m-n)!}[/tex]
where "!" indicates the factorial of a positive integer, which is defined as the product of all natural numbers before or equal to it.
This caseIn this case, you have:
the letter word "PAYMENT", where the number of letters is 7.the letters are taken 6 at a time.Then, you know that:
m= 7n=6Replacing in the definition of permutation:
[tex]7P6=\frac{7!}{(7-6)!}[/tex]
Solving:
[tex]7P6=\frac{7!}{1!}[/tex]
7P6= 5040
Finally, in 5040 ways can the letters in the word ''PAYMENT'' be arranged if the letters are taken 6 at a time.
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The property tax on a house with an assessed value of $624,000 is $7280. Determine the property tax on a house with an assessed value of $762,000, assuming the same tax rate.
Using proportions, the property tax on a house with an assessed value of $762,000 is of $8,890.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
The tax rate, as a proportion, is:
7280/624000 = 0.011667.
Hence, the property tax on a house with an assessed value of $762,000 is of:
0.011667 x 762,000 = $8,890.
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what is the volume of the following rectangular prism
Answer:
[tex] \large{ \boxed{ \tt{336 \: units ^{3} }}}[/tex]- Please see the attached picture for full solution!:)
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a2 + ab -2b2 factorise
[tex]a {}^{2} + ab - 2 {b}^{2} [/tex]
[tex] = (a - b) {}^{2} [/tex]
Find the remainder when f(x) = 8x^3+4x^2-13x+3 is divided by 2x+5
Answer:
-129
Step-by-step explanation:
good luck
A=(-3,-2) B=x,3 C=(4,5) if AB = BC what is the value of X
Answer:
x=1
Step-by-step explanation:
[tex] \sqrt{( - 3 - x) ^{2} + {( - 2 - 3)}^{2} } = \\ \sqrt{ {(4 - x)}^{2} + {(5 - 3)}^{2} } \\ \\ 9 + 6x + {x}^{2} + 25 = \\ 16 - 8x + {x}^{2} + 4 = = = > \\ 34 + 6x = 20 - 8x = > \\ 14 = 14x = = = > x = 1[/tex]
Done
The value of x that makes up the coordinate of B is -1
How to find the value of xTo find the value of x, we need to determine the coordinates of point B.
Since A to (B) = B to (C), the distance from point A to point B is equal to the distance from point B to point C. The distance between two points in a coordinate plane can be calculated using the distance formula:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Let's calculate the distances:
Distance A to (B):
x₁ = -3 (x-coordinate of point A)
y₁ = -2 (y-coordinate of point A)
x₂ = x (x-coordinate of point B)
y₂ = 3 (y-coordinate of point B)
Distance A to (B) = √((x - (-3))² + (3 - (-2))²)
Distance A to (B) =√((x + 3)² + 5²)
Distance A to (B) = √(x² + 6x + 9 + 25)
Distance A to (B) = √(x² + 6x + 34)
Distance B to (C):
x₁ = x (x-coordinate of point B)
y₁ = 3 (y-coordinate of point B)
x₂ = 4 (x-coordinate of point C)
y₂ = 5 (y-coordinate of point C)
Distance B to (C) = √((4 - x)² + (5 - 3)²)
Distance B to (C) = √((4 - x)² + 2²)
Distance B to (C) = √((4 - x)² + 4)
Distance B to (C) = √(x² - 8x + 20)
Since A to (B) = B to (C):
√(x² + 6x + 34) = √(x² - 8x + 20)
Now, we can square both sides to get rid of the square root:
x² + 6x + 34 = x² - 8x + 20
Next, move all the x terms to one side of the equation:
x² - x² + 6x + 8x = 20 - 34
Combine like terms:
14x = -14
Finally, solve for x:
x = -14 / 14
x = -1
So the value of x is -1. Point B is located at (-1, 3).
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A yoga studio charges a $120 yearly fee for membership plus $16 for each class.
What is the average total cost per class if a studio member takes 100 classes per year?
Answer:
$17.2
Step-by-step explanation:
120 + 16*100 = 120 + 1600 = 1720
The member pays $1720 annually for 100 classes and membership fee combined.
1720/100 = 17.2
The student pays $17.2 per class on average.
Please Help Brainliest
Using the z-distribution, the p-value would be of:
b) 0.0086.
What are the hypothesis tested?At the null hypothesis we test if the means are equal, equivalent to a subtraction of 0, hence:
[tex]H_0: \mu_D - \mu_C = 0[/tex]
At the alternative hypothesis, we test if they are different, hence:
[tex]H_1: \mu_D - \mu_C \neq 0[/tex]
What are the mean and the standard error for the distribution of differences?For each sample, they are given as follows:
[tex]\mu_D = 12, s_D = \frac{5.2}{\sqrt{73}} = 0.6086[/tex].[tex]\mu_C = 14, s_C = \frac{4.1}{\sqrt{81}} = 0.4556[/tex].For the distribution of differences, it is given by:
[tex]\overline{x} = \mu_D - \mu_C = 12 - 14 = -2[/tex].[tex]s = \sqrt{s_D^2 + s_C^2} = \sqrt{0.6086^2 + 0.4556^2} = 0.76[/tex]What is the test statistic?The test statistic is given by:
[tex]z = \frac{\overline{x} - \mu}{s}[/tex]
In which [tex]\mu = 0[/tex] is the value tested at the null hypothesis.
Hence:
[tex]z = \frac{\overline{x} - \mu}{s}[/tex]
z = -2/0.76
z = -2.63.
Using a z-distribution calculator, for a two-tailed test, with z = -2.63, the p-value is of 0.0086.
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An object moves in simple harmonic motion with period 6 seconds and amplitude 7cm. At time t=0 seconds, its displacement d from rest is -7cm, and initially it moves in a positive direction.
Give the equation modeling the displacement d as a function of time t.
The equation modeling the displacement 'd' as a function of time is d (t) = 7 sin ( 3π t)
What is simple harmonic motion?
Simple harmonic motion can be defined as a periodic motion of a point in a straight line, such that its acceleration is towards a fixed point and is proportional to its distance from that point to the line.
It is derived from the projection to the axis of a circle of a point in constant speed on the circumference
Displacement is represented thus;
d ( t ) = A sin ( 2 π f t )
T = 1/f
Where,
d is the displacement A is the amplitude = 7cmT is the period = 6 secondsNote that there is no phase shift
d (t) = 7 sin ( 2π × 1/ 6 t)
d (t) = 7 sin ( 2π/6 t)
d (t) = 7 sin ( 3π t)
Thus, the equation modeling the displacement 'd' as a function of time is d (t) = 7 sin ( 3π t)
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Identify a horizontal or vertical stretch or compression of the function f(x) = x
by observing the equation of the function g(x) = 6x.
The base graph vertically stretched by a factor of 6.
The base graph horizontally stretched by a factor of 6.
The base graph vertically compressed by a factor of 6.
The base graph horizontally compressed by a factor of 6.
Answer: The base graph vertically stretched by a factor of 6.
Step-by-step explanation:
See attached image.
The following statements are all incorrect. Explain the statements and the errors fully using the probability rules discussed in topic two. 1.The number 7 is a lucky number so you are more likely to win raffles with ticket number 7 than with a different number. 2.I roll two dice and add the results. The probability of getting a total of 8 is 1/12 because there are 12 different possibilities and 8 is one of them. 3.Mr. Verde has to have a major operation. Ninety-three percent of the people who have this operation make a complete recovery. There is a 93% chance that Mr. Verde will make a complete recovery if he has this operation.4.The Ramblers play the Chargers. The Ramblers can win, loose, or draw, so the probability that they win is 1/3. 5.I have flipped an unbiased coin four times and got heads. It is more likely to get tails the next time I flip it.6.Thirty random college students are asked if they study during the week. Since 60% said yes, a statement can be made that 40% of students only study on the weekend.
The probability is illustrated below.
How to illustrate the probability?1) Raffles game is a gambling competition in which people get numbered tickets and each number has an equal chance of winning.
2) When two dice are rolled then there will be totally 36 outcomes. Of those the outcomes with sum 8 are (2,6),(3,5),(4,4),(5,3),(6,2).
These are 5 in number.
So,P( sum =8)= 5/36.
3)This is a true statement.
4) Here win, lose, or tie is just the outcomes where their probabilities are different from each other and those values depend upon different situations.
5) Tossing a coin is a random experiment which means that the result of the experiment (ie head or tail) cannot be predicted.
6) If 60% said yes to " study during the week" then 40% will say yes to "either won't study or study at weekend".
7) When two coins are tossed, the outcomes are as follows
H,H
H,T
T,H
T,T.
So, P( head, tail ) =2/4 =1/2
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The reasons for each incorrect statement are as follows;
The error in the statement is that, there might be other luckier numbers compared to number 7 and hence, the statement does not hold true for all numbers.The statement is incorrect because, there are 36 different possibilities and there are 5 possibilities of getting an 8.The statement is incorrect because, 93% does not necessarily represent a 93% chance of complete recovery as this is subject to other variables.The probability that they win is not 1/3 as it is subject to what the results have been over the years and not just the possible results.Since, the coin is unbiased, it follows that upon flipping the coin for the fifth time, it is not likely to have tails the next time it is flipped.The statement is incorrect because some students study during the week and weekends, hence, the conclusion is not correct.What is Probability?The term probability put simply is the ratio of the desired outcome to the possible outcomes. It therefore follows that probability as the name implies is used to predict the chance of an occurrence.
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An administer of a college believes that 40% students of the college work in part time. He
randomly surveyed of 2,512 students and finds that 1,298 of them work in part time, that is, about 51.7% of
the student being surveyed work in part time.
1.
Is the study observational or experimental?
2.
What is the population of interest?
3.
What is the parameter of interest?
4.
What is the sample in the study?
5.
What is the statistic involved?
See below for the response to each question
Is the study observational or experimental?The study is experimental.
This is so because, the administer surveyed random college students
The population of interestThis is the total number of students in the college
Hence, the population of interest is the students in the college
The parameter of interestThis is the value that gives the required information.
Hence, the parameter of interest is the proportion of college students that work part-time
The sample in the studyThis is the students that were randomly surveyed
Hence, the sample in the study is the 2,512 students that were randomly surveyed
What is the statistic involvedThe statistic is the sample mean
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a group of students was randomly divided into two subgroups. one subgroup took a test in the morning, and the other took the same test in the afternoon. this table shows the results. compare the probability that a student will pass the test in the morning with the probability that a student will pass the test in the afternoon. draw a conclusion based on your results.
The conclusion is P(pass l morning) = 0.56
P(pass l afternoon) = 0.72
Conclusion: a student taking the test in the afternoon has a greater probability of passing than a student taking the test in the morning.
What are the probabilities?
Probability is the odds that a random event would happen. The odds that the event would happen lies between 0 and 1. The closer the probability is to one, the more likely it is for the event to happen.
P(pass l morning) = number of students who took the test in the morning and passed / total number of students that took the exam in the morning
28 / 50 = 0.56
P(pass l afternoon) = number of students who took the test in the afternoon and passed / total number of students that took the exam in the afternoon = 36 / 50 = 0.72
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Answer:
C
Step-by-step explanation:
Just took the quiz
Match the information on the left with the appropriate equation on the right.
m=1, b=-3. x+y=1
m=1, (-1,2) x-y=-1
(-2,3),(-3,4) x-y=-3
x-y=3
The equation of the line passing the points (-2,3),(-3,4) is y = -x + 1 or x+y = 1
Equation of a lineA line is the shortest distance between two points. The equation of a line in slope-intercept form is expressed as;
y = mx + b
m is the slope
b is the y-intercept
Given that m=1, b=-3, the required equation will be y = x - 3 or x - y = 3
For the parameter m=1, (-1,2)
y - 2 =1(x+1)
y - 2 = x+1
y =x + 3
Hence the equation of the line for the parameters m=1, (-1,2) is y = x + 3
For the coordinates (-2,3),(-3,4)
Slope = 4-3/-3+2
Slope = -1
For the intercept;
4 =-1(-3) + b
b = 4 - 3
b = 1
Hence the equation of the line passing the points (-2,3),(-3,4) is y = -x + 1 or x+y = 1
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In 2003, Dullinger et al. examined patterns of shrub invasion into high mountain grasslands of the Northern Calcareous (Limestone) Alps in Austria. Dullinger and his colleagues wished to determine whether Pinus mugo, a species of invasive conifer, showed a preference for invading some grassland habitats over others. The data provided give the number of Pinus mugo observed in each of six grassland types studied by the investigators. Assume that the area covered by each of the six habitats studied are approximately equal. What is the Pvalue?
based on the given area for the six habitats, Dullinger and his colleagues can find the p-value to be 0.0018.
how can the p-value be found?a chi-squared test is the best test for to find the p-value and when the data given is put through a chi-square model, the observed values are:
= 7, 9, 2, 11, 2, 0
the expected values should be:
= 1/6, 1/6, 1/6, 1/6, 1/6, 1/6
the p-value will be 0.001819
df = 5
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Need help with this question
The domain f the function is (-∞, 1/3) U (1/3, ∞)
Composite functionsThe composite functions are also known as function of a function
Given the following functions
f(x) = √2-6x
g(x) = -1/x
In order to determine the composite function g(f(x))
g(f(x)) = g(√2-6x)
Replace x with √2-6x in g(x)
g(f(x)) = -1/√2-6x
The expression does not exist when √2-6x = 0
2-6x = 0
x = 1/3
The values shows that the domain of the function exists on all real numbers except 1/3.
Hence the domain f the function is (-∞, 1/3) U (1/3, ∞)
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ped
Complete question :
Juan is rewriting the expression 12 d minus 26 c as a product. Which statements about the expression are accurate and relevant to his rewriting the expression? Select two options. The GCF of the numbers in each term in the expression is 2. The GCF of the numbers in each term in th
So the correct statement is:
" The GCF of the numbers in each term in the expression is 2"
How to rewrite the expression as a product?Here we have the expression:
12d - 26c
To rewrite ti as a product, we need to find the greatest common factor between 12 and 26.
The decomposition of these two numbers is:
12 = 2*2*3
26 = 2*13
The greatest common factor between the two numbers is 2, then we can rewrite:
12d - 26c = 2*(6d - 13c)
So the correct statement is:
" The GCF of the numbers in each term in the expression is 2"
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Example 2.11: Solve the following
a) 4x² +10x = 6
Answer:
[tex]\huge\boxed{\sf \{-3, 1/2\}}[/tex]
Step-by-step explanation:
Given equation:4x² + 10x = 6
Take 2 common
2(2x² + 5x) = 6
Divide 2 to both sides
2x² + 5x = 3
Subtract 3 to both sides
2x² + 5x - 3 = 0
Using mid-term break
2x² + 6x - x - 3 = 0
Take common
2x(x + 3) - 1(x + 3) = 0
Take (x + 3) common
(x + 3)(2x - 1) = 0
Either,
x + 3 = 0 OR 2x - 1 = 0
x = -3 OR 2x = 1
x = -3 OR x = 1/2
Solution Set = {-3, 1/2}[tex]\rule[225]{225}{2}[/tex]
Answer: x = 3 or (-1/2)
Step-by-step explanation:
Given equation
4x² + 10x = 6
Subtract 6 on both sides
4x² + 10x - 6 = 6 - 6
4x² + 10x - 6 = 0
Use cross-multiplication to factorize the quadratic
Starting with:
2x -6
2x 1
This gives us:
(2x - 6) (2x + 1) = 0
Assume each value in the parenthesis to be 0
PART I:
2x - 6 = 0 ⇒ Given equation
2x - 6 + 6 = 0 + 6 ⇒ Add 6 on both sides
2x / 2 = 6 / 2 ⇒ Divide 2 on both sides
[tex]\boxed{x=3}[/tex]
PART II:
2x + 1 = 0 ⇒ Given equation
2x + 1 - 1 = 0 - 1 ⇒ Subtract 1 on both sides
2x / 2 = -1 / 2 ⇒ Divide 2 on both sides
[tex]\boxed{x=-\frac{1}{2} }[/tex]
Hope this helps!! :)
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Physics shows that when an object is thrown upward with an initial velocity of v then its approximate height v_o is given by this quadratic function.
s = − 4.9t^2 + v_ot + h
A ball is thrown upward from the top of a building 15 meters high, at a velocity of 2.8m/sec. Answer the following questions. Be sure to show and explain all work.
a) Find the maximum height and the time in which it was attained (round to 3 decimal places)
b) When does it reach the ground? (round to 3 decimal places)
c) Sketch a graph to illustrate your answers.
The maximum height of the object is 15.4 m and occurs at 0.295 sec. The object touches ground at 2.06 seconds
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
A ball is thrown upward from the top of a building 15 meters high, at a velocity of 2.8m/sec. Hence:
s = -4.9t² + 2.8t + 15
The maximum height is at s'(t) = 0, hence:
-9.8t + 2.8 = 0
t = 0.295 s
s = -4.9(0.295)² + 2.8(0.295) + 15 = 15.4 m
The object reaches ground when s = 0, hence:
0 = -4.9t² + 2.8t + 15
t = 2.06 seconds
The maximum height of the object is 15.4 m and occurs at 0.295 sec. The object touches ground at 2.06 seconds
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I forgot how to find the range of values of x and I will be very much appreciated if I could get a step by step clear solution to get the answers!
The quadratic equation has the following results: (p, q) = (8, 15), minimum point: (h, k) = (1, 4), range of values: - 4 < x < - 3
How to analyze quadratic equations
In this question we have a quadratic equation of the form y = x² + p · x + q , whose missing coefficients can be found by solving on the following system of linear equations:
- 5 · p + q = - 25
- 3 · p + q = - 9
(p, q) = (8, 15)
The vertex represents the minimum point, which is found by changing the form of the equation from standard form into vertex form:
y = x² + 8 · x + 15
y + 1 = x² + 8 · x + 16
y + 1 = (x + 4)²
(h, k) = (1, 4)
And lastly we must solve for x in the following inequality:
x² + 8 · x + 15 < x + 3
x² + 7 · x + 12 < 0
(x + 3) · (x + 4) < 0
- 4 < x < - 3
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IM IN A HURRY PLEASE HELP ME QUESTION IS DOWN BELOW WORTH 15 POINTS each
a. central angle is ∠QPR
b. semicircle is QRS
c. major arc is arc QTS
d. m∠QPR = 130° [central angle theorem]
e. arc QTS = 180°
What is the Central Angle Theorem?The central angle theorem states that the measure of the intercepted arc will always be equal to the central angle.
a. A central angle is ∠QPR
b. A semicircle is QRS
c. A major arc is arc QTS
d. Measure of arc QR = m∠QPR = 130° [central angle theorem]
e. arc QTS = semicircle = 180°
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The bearing from the Pine Knob fire tower to the Colt Station fire tower is N 65° E, and the two towers are 31 kilometers apart. A fire spotted by rangers in each tower has a bearing of N 80° E from the Pine Knob and S 70° E from Colt Station (see figure). Find the distance of the fire from each tower. (Round your answers to two decimal places.)
From Pine Knob:
The distance are
a. 42.42
b. 15. 52
How to solve for the distance80 = 65 + <ABC
<ABC = 80 - 65
= 15
Through the use of sine rule we would have
By using sine rule,
Sine 15/a = sine 135/b =
sin 30/30.
Next we have to solve b by cross multiplying
b = (30× sin135) / sin 30
= 21.21/0.5
b = 42.42 km
Also, the value of a will be:
= (42.42 × sin 15) / sin 135
= 10.98/0.7071
= 15.52
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help guys please. I
Step-by-step explanation:
you need to use the ruler to measure the lengths of it, like place the ruler from A to B and look at the measurement of it
Enter the letter of the graphed
function.
-
-
a.y = (x + 2)(x − 1)(x − 3)
b.y = (x + 2)(x - 1)(x - 3)²
-
c. y = (x + 2)(x − 3)(x - 1)²
d.y= (x - 1)(x − 3)(x + 2)²
e.y = (x - 1)(x - 3)(x + 2)³
The graphed function is: y = (x + 2)(x − 1)(x − 3).
Option (a) is correct.
From the graph given in the question, it can be seen that the graph has 3 critical points, namely, x= -2, x = 1 and x = 3.
From all the five options given, the first option, that is, y = (x + 2)(x − 1)(x − 3), has the critical points x= -2, x = 1 and x = 3.
The critical points for a function are obtained by substituting the function to be equal to zero.
So, (x + 2)(x − 1)(x − 3) = 0
That is, x= -2, x = 1 and x = 3.
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Josh had to make 2,000 cookies for his bakery. 2 pieces of wheat make 8 cookies. How much wheat does he need?
explain why the equation y=5x is a special case of the slope-intercept form. what are the slope a nd y intercept of the line represented in this equation
Answer:
It is proportional. When You graph it, it will go through the origin. The slope is 5. The y intercept is 0.
Step-by-step explanation:
When a linear function is proportional, the slope will be in the form of y/x.
So, say we put 2 in for x, then y would be 10. 2 is the x and 10 is the y. What is 10/2? 5 (the slope)
If we put in 6 for x, then y would be 30. 5 is the x and 30 would be the y. What is 30/5? 5 (the slope)
if we put in -3 for x, then y would be -15. -3 is the x and -15 would be the y. What is -15/-3? 5 (the slope).
We could not find the slope this way, if the line was not proportional.
For example, if the y intercept is not zero, like
y = 5x + 2. If I put 3 in for the x, then the y would be 17. Is 17/3 equal to 5? No, because the line is not proportional.
A bag contains five yellow tickets numbered one to five. The bag also contains five green tickets numbered one to five. You randomly pick a ticket. It is green or has a number greater than four. Find the probability of this occuring.
The probability of picking a ticket that is green or has a number greater than four is 3/5
How to determine the probability?The given parameters are:
Yellow = 1 - 5
Green = 1 - 5
Total = 10
There are 2 cards whose numbers are greater than 4 i.e. Yellow 5 and Green 5
So, we have:
P(Number greater than 4) = 2/10
There are 5 green cards.
So, we have:
P(Green) = 5/10
Also, 1 green card is numbered greater than 4
So, we have:
P(Green greater than 4) = 1/10
The required probability is:
P = P(Green) + P(Number greater than 4) - P(Green greater than 4)
This gives
P = 5/10 + 2/10 - 1/10
Evaluate
P = 6/10
Simplify
P =3/5
Hence, the probability of picking a ticket that is green or has a number greater than four is 3/5
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Initially, Ernie does not like a book that he is reading, so he reads very few pages. Then, the story becomes interesting, and he reads more pages. The more Ernie reads, the more he enjoys, so he reads faster and faster until he finishes the book. Choose the graph that best corresponds to the situation.
The graph that best corresponds to the situation is graph 4; option C
Time graphAssume, the first hour, Ernie read just 2 pagesSecond hour, 5 pagesThird hour, 7 pagesFourth hour, 10 pagesNumber of pages read is on the y - axis
Time taken is on the x - axis
Therefore, graph 4 represent the situation
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consider function g. g(x) = { (1/2)^x +3, x< 0. -x^2 +2, x>_ 0
[tex]\square[/tex] The function is continuous. [False]
Both pieces of the function are continuous, so the overall continuity of [tex]g(x)[/tex] depends on continuity at [tex]x=0[/tex].
We have
[tex]\displaystyle \lim_{x\to0^-} g(x) = \lim_{x\to0} \left(\frac1{2^x} + 3\right) = 1 + 3 = 4[/tex]
and
[tex]\displaystyle \lim_{x\to0^+} g(x) = \lim_{x\to0} (-x^2+2) = 2[/tex]
The one-sided limits do not match, so [tex]g[/tex] is not continuous at [tex]x=0[/tex].
[tex]\square[/tex] As [tex]x[/tex] approaches positive infinity, [tex]g(x)[/tex] approaches positive infinity. [False]
[tex]g(x)[/tex] is a large negative number when [tex]x[/tex] is very large, so [tex]g(x)[/tex] is approaching negative infinity.
[tex]\boxed{\checkmark}[/tex] The function is decreasing over its entire domain. [True]
This requires [tex]g'(x) \le 0[/tex] on the entire real line. Compute the derivative of [tex]g[/tex].
[tex]g'(x) = \begin{cases}-\ln(2)\left(\dfrac12\right)^x & x<0 \\\\ ? & x=0 \\\\ -2x & x>0 \end{cases}[/tex]
• [tex]\left(\frac12\right)^x > 0[/tex] for all real [tex]x[/tex], so [tex]g'(x)<0[/tex] whenever [tex]x<0[/tex].
• [tex]x^2\ge0[/tex] for all real [tex]x[/tex], so [tex]-x^2\le0[/tex] and [tex]-x^2+2\le2[/tex]. Equality occurs only for [tex]x=0[/tex], which does not belong to [tex]x>0[/tex].
Whether the derivative at [tex]x=0[/tex] exists or not is actually irrelevant. The point is that [tex]g(b) < g(a)[/tex] if [tex]b>a[/tex] for all real [tex]a,b[/tex].
[tex]\boxed{\checkmark}[/tex] The domain is all real numbers. [True]
There are no infinite/nonremovable discontinuities, so all good here.
[tex]\boxed{\checkmark}[/tex] The [tex]y[/tex]-intercept is 2. [True]
When [tex]x=0[/tex],
[tex]g(0) = -0^2 + 2 = 2[/tex]