One way to show a function is 1-1 is to show that if two outputs are the same, then the inputs had to have been the same as well:
If f(a) = f(b), then a=b.
If you can show this, you've shown that each output is paired with a unique input, so the function is 1-1.
For this function, we'd first start assuming that f(a) = f(b):
f(a) = f(b)
-3a+5 = -3b+5
Now we'll solve this for a:
-3a + 5 = -3b + 5
-3a + 5 - 5 = -3b + 5 - 5
-3a = -3b
-3a ÷ (-3) = -3b ÷ (-3)
a = b
This shows that the only way the outputs are the same is if the inputs are the same, which means each output comes from a unique input, so the function is 1-1.
James exercised for 2/5 of an hour and sandile exercised for 4/12 of an hour ? Who exercised for the longest time?
Answer:
The answer James
Step-by-step explanation:
James=2/5hours
J=2/5×60=2×12=24 minutes
Sandle=4/12×60=4×5=20 minutes
Solve This Mathematical Term
The reason of discontinuity of the given function is described below.
Since we know that,
A continuous function, as the name implies, is one whose graph is continuous with no breaks or jumps. In other words, if we can draw the curve (graph) of a function without ever raising a pencil, we may claim that the function is continuous. The study of a function's continuity is critical in calculus because a function cannot be differentiable unless it is continuous.
A function is considered continuous if its graph is an uninterrupted curve with no holes, gaps, or breaks.
Now by mathematical definition of continuity
Left hand limit of a function at any point = right hand limit of function at that point = value of function at that point
Now from the graph we can see that,
At point point 3
The left hand limit and right hand limit are not equal and also the curve is discontinuous.
Hence, the given function is discontinuous function.
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SP - Rs 325 profit percent - 25% find Cp
⌢
In circle D, m∠EDF=70∘, and the length of EF=[tex]\frac{14}{9} \pi[/tex]. Find the length of DE.
To find the length of DE in circle D, we can use the properties of a circle and the given information.
In a circle, if two chords intersect, the product of the segments of one chord is equal to the product of the segments of the other chord. In this case, we have chord EF intersecting chord DE at point D.
Let x represent the length of segment DE, and y represent the length of segment DF. Therefore, the length of segment EF would be (x + y).
According to the chord-chord power theorem:
DE * EF = DF * DF
Substituting the given values:
x * (x + y) = y * y
x^2 + xy = y^2
We are also given that angle EDF measures 70 degrees. According to the angle intercepting chord theorem, the intercepted arc EF is twice the measure of angle EDF. So, the measure of arc EF is 2 * 70 = 140 degrees.
Now, we can use the length of arc EF to find the ratio of the lengths of segments EF and DF.
The ratio of the lengths of the intercepted arcs is equal to the ratio of the lengths of the corresponding chords. Therefore:
EF / DF = arc EF / arc DF
(x + y) / y = 140 / 360 [Using the measure of the intercepted arcs]
Simplifying this equation:
(x + y) / y = 7 / 18
Cross-multiplying:
18(x + y) = 7y
18x + 18y = 7y
18x = 7y - 18y
18x = -11y
Dividing by -11:
x = -11y / 18
We need to find the value of x, which represents the length of segment DE. Since segment lengths cannot be negative, we can disregard the negative sign.
x = 11y / 18
Substituting this value of x in the equation x^2 + xy = y^2:
(11y / 18)^2 + (11y / 18) * y = y^2
Simplifying:
121y^2 / 324 + 11y^2 / 18 = y^2
Multiplying through by 324:
121y^2 + 594y^2 = 324y^2
715y^2 = 324y^2
715y^2 - 324y^2 = 0
391y^2 = 0
y^2 = 0
Since y^2 = 0, it implies that y = 0. This means that segment DF has zero length.
Now, substituting y = 0 into the equation x = 11y / 18:
x = 11 * 0 / 18
x = 0
Therefore, the length of DE is 0.
In conclusion, the length of DE is 0.
I cannot load the image that you have provided this is what the answer is according to the text provided.
Twenty-five psychology instructors have formed a committee to pick next year's textbook, and they have narrowed their decision down to two equally good books, one with a better bibliography and references, and the other with a better format and illustrations. Since the books are considered to be equally good, we will assume the probability an instructor chooses either book is 0.5 and the instructors' decisions are made independently. Using the binomial distribution, find the probability 15 or more instructors choose the book with the better format and illustrations.
To find the probability that 15 or more instructors choose the book with the better format and illustrations, we can use the binomial distribution formula.
Let's denote the event of an instructor choosing the book with the better format and illustrations as "success" (S), and the event of an instructor choosing the other book as "failure" (F). The probability of success is p = 0.5, and the probability of failure is q = 1 - p = 0.5.
We want to find the probability of 15 or more successes in a sample of 25 instructors. We can calculate this probability by summing the probabilities of exactly 15, 16, 17, ..., 25 successes.
P(X ≥ 15) = P(X = 15) + P(X = 16) + ... + P(X = 25)
Using the binomial distribution formula, the probability of exactly k successes in a sample of n trials is:
P(X = k) = C(n, k) * p^k * q^(n-k)
where C(n, k) is the binomial coefficient "n choose k," given by:
C(n, k) = n! / (k! * (n-k)!)
Applying this to our problem, we can calculate the probability as follows:
P(X ≥ 15) = P(X = 15) + P(X = 16) + ... + P(X = 25)
= Σ[ k = 15 to 25 ] ( C(25, k) * p^k * q^(25-k) )
Let's calculate this probability using the binomial distribution formula:
P(X ≥ 15) = Σ[ k = 15 to 25 ] ( C(25, k) * (0.5)^k * (0.5)^(25-k) )
Calculating this sum gives us the probability that 15 or more instructors choose the book with the better format and illustrations.
of the 150 pupils in grade 7 at mpenzeni primary school, 90 passed the final examination in 2015. what percentage of the pupils pass
Given statement solution is :- 60% of the pupils in grade 7 at Mpenzeni Primary School Pass Rate the final examination in 2015.
To find the percentage of pupils who passed the final examination, you can use the following formula:
Percentage = (Number of pupils passed / Total number of pupils) * 100
Given that 90 pupils passed the final examination out of a total of 150 pupils, we can substitute these values into the formula:
Percentage = (90 / 150) * 100
Percentage = 0.6 * 100
Percentage = 60%
Therefore, 60% of the pupils in grade 7 at Mpenzeni Primary School Pass Rate the final examination in 2015.
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Calculate the surface area of the triangular prism
Answer:
66 inches²
Step-by-step explanation:
Area of 1 triangle = ½ × base × height = ½ × 7 × 4 = 14 in²
Area of 2 triangles = 14 × 2 = 28 in²
Area of outer rectangles = (6 × 2) × 2 = 24 in²
Area of inner rectangle = 2 × 7 = 14 in²
Total surface area = 28+24+14 = 66 in²
Question 5 of 10
What can you say about the y-values of the two functions f(x) = 32 - 3
and g(x) = 7x² - 3?
A. The minimum y-value of f(x) is -3.
B. The minimum y-value of g(x) is -3.
C. g(x) has the smallest possible y-value.
D. f(x) has the smallest possible y value.
SUBMIT
The minimum y-value of g(x) is -3 and g(x) has the smallest possible y-value.
The function f(x) = 32 - 3 is a constant function, as there is no variable term involving x. It simplifies to f(x) = 29.
Regardless of the value of x, the function f(x) will always have a y-value of 29.
Therefore, option A is incorrect because there is no minimum y-value for f(x); it is a constant.
g(x) = 7x² - 3 is a quadratic function in the form of ax² + bx + c, where a = 7, b = 0, and c = -3.
The coefficient a (7) is positive, indicating that the parabola opens upward.
Since there is no linear term (bx), the vertex of the parabola is located at the point (0, -3), which corresponds to the minimum y-value of the function.
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Question 1 of 10
In the ellipse shown below, each point marked with a dot is called a(n)
A. vertex
B. axis
O c. center
OD. focus
In the ellipse shown below, each point marked with a dot is called a(n) focus
What is a focus in an ellipseIn an ellipse, the focus (plural: foci) is a pair of special points that play a significant role in defining the shape of the ellipse. The focus points are located inside the ellipse along the major axis, which is the longer axis of the ellipse.
The defining property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant. This property is known as the "focus-directrix property."
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Find the inverse function Y = x² for x>0
Answer:
y= √x--------------
Given function:
y = x²In order to find its inverse, first swap x and y:
y = x² ⇒ x = y²Then solve for y:
y² = xy = √xThe inverse function is y= √x.
pls solve and explain
Answer:
[tex]\theta = 60; \theta \approx 83; \theta \approx 277; \theta = 300[/tex]
Note that the approximate value you should calculate properly with the inverse cosine function of your calculator to the number of digits you want.
Step-by-step explanation:
you already have a cosine. You might as well change that sine in a cosine too:
[tex]8(1-cos^2\theta) +6cos\theta-9=0\\8cos^2\theta -6cos\theta+1 =0[/tex]
(note that in the second line I also switched every sign by multiplying by -1 just to have a positive lead coefficient - just a pet peeve of mine. Now replace [tex]x=cos\theta[/tex] and let's solve
[tex]8x^2-6x+1=0\\\frac\Delta4 = 9-8 = 1\\x=\frac {3\pm1}8[/tex]
At this point we're back to the replacement. We have two options
[tex]cos\theta = \frac48 = \frac 12 \implies \theta = 60\°; \theta = 300\°\\cos\theta =\frac 18 \implies \theta \approx 83\°; \theta \approx 277\°[/tex]
In ΔNOP, p = 70 inches, n = 97 inches and ∠O=163°. Find ∠N, to the nearest degree.
Answer:
10°----------------------
In ΔNOP, we are given:
p = 70 inches, n = 97 inches, ∠O = 163°Use the law of cosines to find side o:
o² = n² + p² - 2(np)cos(∠O) o² = 97² + 70² - 2(97)(70)cos(163°) o² = 27295.61o = 165.2Now, use the law of sines to find ∠N:
sin(∠N) / n = sin(∠O) / o sin(∠N) / 97 = sin(163°) / 165.2sin(∠N) = 97*sin(163°) / 165.2sin(∠N) = 0.17m∠N = arcsin (0.17)m∠N = 9.78° ≈ 10°So, ∠N is approximately 10°.
In a village of 320 adults, each of them does at least farming or trading. 5/8 of them do farming and 75% of these who do farming are involved in trading. a. Find the number of people who are involved in: i. Farming ii. Trading. b. How many of the people are involved in exactly two activities?
c. Find the number of people who do only one activity.
d. What percentage of the people are traders only?
e. What is the probability of selecting an adult from the village at random who is a farmer but not a trader?
Answer:
please see answers below
Step-by-step explanation:
i) 5/8 X 320 = 200 people
ii) 0.75 X 200 = 150 (people in farming but also trade). only trade = 320 - 200 = 120.
number of people in trade = 150 + 120 = 270.
b) 0.75 X 5/8 = 3/4 X 5/8 = 15/32 = 150 people (both farm and trade).
c) only farming = 0.25 X 5/8 = 1/4 X 5/8 = 5/32 = 50.
only trade = 320 - 50 (only farm) - 150 (both farm and trade) = 120.
so, only do one activity = 50 + 120 = 170.
d) traders only = 120/320 = 3/8 = 37.5%
e) P(farmer only) = 50/320 = 0.15625
how would I solve this
[tex]\theta =\cfrac{14}{3}\pi \implies \theta =\cfrac{14\pi }{3}\implies \theta =\cfrac{6\pi +6\pi +2\pi }{3} \\\\\\ \theta =\cfrac{6\pi }{3}+\cfrac{6\pi }{3}+\cfrac{2\pi }{3}\implies \theta =2\pi +2\pi +\cfrac{2\pi }{3}[/tex]
so if we take a look at that, we can say that the angle θ does two revolutions and then it lands on 2π/3, so the terminal point of it is the same as 2π/3's, and if you check your Unit Circle, as you should have one
[tex]\sin(\theta )=\cfrac{\sqrt{3}}{2}\hspace{5em}\cos(\theta )=-\cfrac{1}{2}[/tex]
Answer:
sin∅ = √3/2
cos∅ = -1/2
Step-by-step explanation:
Notice that 14π/3 is just under 3π, so the reference angle made with the negative x-axis is π/3. Since sin(π/3) = √3/2, then sin(14π/3) is also √3/2.
cos(14π/3) works somewhat similarly. Since cos(π/3) = 0.5, and cos(14π/3) would be located in Quadrant II where the negative axis is, then cos(14π/3) = -0.5
Jaxon is flying a kite, holding his hands a distance of 3.25 feet above the ground and letting all the kite’s string play out. He measures the angle of elevation from his hand to the kite to be 24
∘
∘
. If the string from the kite to his hand is 105 feet long, how many feet is the kite above the ground? Round your answer to the nearest tenth of a foot if necessary.
MP Precision Three students measure the height of a bookshelf. Student A measures 72 units, Student B measures units, and Student C measures 6 units. The teacher says all three students are correct. What units did each student use? 15
The units that each student used, given the measurement that they got and all of them being right was:
Student A used inchesStudent B used feetStudent C used yards.How to find the units used ?The height of a bookshelf is typically measured in inches, feet, or yards .
Student A measured 72 units, which is equal to 6 feet. Student B measured 2 units, which is equal to 24 inches .
Student C measured 6 units, which is equal to 18 yards. All three students are correct, as they all used different units to measure the same height .
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Full question is:
Three students measure the height of a bookshelf. Student A measures 72 units, Student B measures 2 units, and Student C measures 6 units. The teacher says all three students are correct. What units did each student use?
a) Simplify p x q x 4
b) Simplify t x t x t
The expressions are simplified to;
a. 4pq
b. t³
What are algebraic expressions?Algebraic expressions are defined as expressions that are made up of terms, variables, coefficients, factors and constants.
These algebraic expressions are also made up of mathematical or arithmetic operations, such as;
AdditionMutiplicationDivisionBracketParenthesesSubtractionFrom the information given, we have that;
The expressions are;
1. p x q x 4
Multiply the terms, we have that;
4qp
2.t x t x t
Multiply the terms, we get;
t³
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why two +Eleven + nine
Answer:
22 is answer
Step-by-step explanation:
Answer:
лвоалалалоала.сллслала лвлвлвл
1 Assignment find p what I = #192 R=2% T = 4years
Answer:
15.36
Step-by-step explanation:
192*0.02*4= 15.36
Inequalities - Introduction
Answer:
Step-by-step explanation:
This number line says that we can take all values of x that are less than or equal to 2. That is, the solution is
[tex]x\leq 2[/tex]
NOTE: if the circle at 2 were not filled in then that would mean x<2 (2 would not be an accepted value of x)
need help asp please
Answer:
see explanation
Step-by-step explanation:
using any of the tangent, sine, cosine ratios in the right triangle.
tanΘ = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{7}{6}[/tex] , then
Θ = [tex]tan^{-1}[/tex] ( [tex]\frac{7}{6}[/tex] ) ≈ 49.40° ( to 2 decimal places )
-------------------------------------------------------------------
sinΘ = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{2}{6.32}[/tex]
cosΘ = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{6}{6.32}[/tex]
tanΘ = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{2}{6}[/tex]
The graph below shows the solution to which system of inequalities?
About 99.7 percent of the monthly rental are between 400 and 430
Answer: rental is $415, and the standard deviation is $5.
Step-by-step explanation:
If about 99.7 percent of the monthly rentals fall between 400 and 430, it implies that this range encompasses three standard deviations from the mean.
To calculate the mean, we can find the midpoint of the given range:
Mean = (400 + 430) / 2 = 415
Since the range of three standard deviations covers about 99.7 percent of the data in a normal distribution, we can use this information to estimate the standard deviation (σ).
Standard deviation (σ) = (430 - 415) / 3 = 15 / 3 = 5
Therefore, based on the given information, the mean monthly rental is $415, and the standard deviation is $5.
HELP! Can someone please help with the picture question below?
The probability that either event A or B will occur is 11/17.
We have,
The Addition Rule for Probability:
P(A U B) = P(A) + P(B) - P(A ∩ B),
where P(A U B) represents the probability of the union of events A and B, P(A) represents the probability of event A, P(B) represents the probability of event B, and P(A ∩ B) represents the probability of the intersection of events A and B.
Now,
P(A) = 9/17
P(B) = 2/17
There is no intersection of Q and B so,
P(A ∩ B) = 0
Now,
P(A U B) = P(A) + P(B) - P(A ∩ B),
Substituting the values.
P(A U B) = The probability that either event A or B will occur.
So,
P(A U B)
= 9/17 + 2/17 - 0
= 11/17
Thus,
The probability that either event A or B will occur is 11/17.
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Rewrite the function by completing the square.
h (x)=x^2+3x−18
Answer: (x+6)(x-3)
Step-by-step explanation:
y=x^2+3x-18
(x+6)(x-3 )
A man had a square window with sides equal to 1 metre in his house. He decided that it lets in too much light, so he boarded up half of it and still had a square window 1 metre high and 1 metre wide. How did he do that? (6) Total: 40 marks
The man rotated the window by 45 degrees and covered one of the diagonals, resulting in a smaller square window with sides measuring 1 meter.
The man achieved the transformation of his square window by dividing it into two equal halves and boarding up one of them.
Let's analyze the steps he took to accomplish this:
Initially, the man had a square window with sides measuring 1 meter
This means that the window had an area of 1 square meter (1m x 1m = 1m²).
To reduce the amount of light coming through the window, the man decided to board up half of it.
This implies that he covered one of the equal halves of the square window, while leaving the other half open.
By boarding up half of the window, he effectively blocked off half of the area of the square window.
Since the original window had an area of 1 square meter, boarding up half of it reduces the effective area to 1/2 square meter (1m² ÷ 2 = 1/2m²).
The remaining open half of the window still retains its square shape, with sides equal to 1 meter.
Therefore, the man ends up with a square window that measures 1 meter in height and 1 meter in width, but with an area of only 1/2 square meter.
In summary, the man achieved the transformation by boarding up half of the square window, effectively reducing its area to half of the original size while maintaining the same dimensions for the remaining open half. This manipulation allows him to control the amount of light entering his house while preserving the square shape of the window.
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Topic: geometry
In the photo
1)
The trigonometric functions :
sinA = 5/13 , cosA = 13/12 , tanA = 5/13
Given,
Right angled triangle with:
P = 5
B = 12
H = 13
Then trigonometric ratios,
sinA = P/H
cosA = B/H
tanA = P/B
Hence the ratios can be defined as,
sinA = 5/13
cosA = 12/13
tanA = 5/12
2)
The value x in the radical form 10.77
Given,
Right angled triangle,
Perpendicular = 4
Base = 10
Hypotenuse = x
Apply pythagora's theorem,
P² + B² = H²
4² + 10² = H²
16 + 100 = H²
H = 10.77
Hence the value of x is 10.77 .
3)
The value of x in radical form
Given,
P = 56
B = x
H = 106
Apply pythagora's theorem,
P² + B² = H²
56² + x² = 106²
x = 90
Hence the value of x is 90 .
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The cost of 1kg potatoes and 2kg tomatoes was 30 on a certain day. After two days the cost of 2kg potatoes and 4kg tomatoes was found to be 66.
expanded form of 73.09
Answer: 70 + 3 + [tex]\frac{9}{100}[/tex]
Step-by-step explanation:
Firstly, what is expanded form? Expanded form is a sum of a number's digit place values.
For example, the expanded form of 814 is 800 + 10 + 4.
To find the expanded form of 73.09, we will do the same as the example provided above. However, we have a decimal. This means we will use a fraction. This 9 is in the hundredth place, so we will use the fraction [tex]\frac{9}{100}[/tex].
73.09 = 70 + 3 + [tex]\frac{9}{100}[/tex]
Express y in terms of x if Log 10 x + Log 10 ( Y )= 2 Log 10 (x+1)
