Suppose that the function f is defined, for all real numbers, as follows.
f(x)= {5x-4 if x≤1
-x+4 if x>1
Graph the function f. Then determine whether or not the function is continuous.
Answers
Answer:
F(x)
Step-by-step explanation:
hope it helps
Given the function [tex]f(x) = \left \{ {{5x-4, \ \ \ x\leq 1} \atop {-x+4, \ \ \ \ \ x > 1}} \right.[/tex], the function is not continuous at x = 1.
A function is said to be continuous if it can be drawn without picking up the pencil. Otherwise, a function is said to be discontinuous.
Given function,
[tex]f(x) = \left \{ {{5x-4, \ \ \ x\leq 1} \atop {-x+4, \ \ \ \ \ x > 1}} \right.[/tex]
We need to check the continuity of the function f(x) at x = 1. At all other points, it is continuous.
f(x) at x = 1 : 5x - 4 = 5*1 - 4 = 5 - 4 = 1
f(x) at x < 1 : 5x - 4 = 5*1 - 4 = 5 - 4 = 1
f(x) at x > 1 : -x + 4 = -1 + 4 = 3
Since, f(x) at x < 1 ≠ f(x) at x > 1, therefore, f(x) is not continuous at x = 1.
The same is also visible from the graph attached below. There is a disruption at x = 1, the graph is not a smooth line.
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Related Questions
Which of the following describes the graph of the
equation 3y = 6x +12?
a line with slope 2 and y-intercept (0,4)
a line with slope 2 and y-intercept (0, 12)
a line with slope 6 and y-intercept (0,4)
a line with slope 6 and y-intercept (0, 12)
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First write the equation in the form y=mx+c
[tex] \frac{3y}{3} = \frac{6x}{3} + \frac{12}{3} \\ y = 2x + 4[/tex]
YOU CAN SEE THAT NOW THE SLOPE OF THE LINE IS 2 TO GET THW Y INTERCEPT WE KNOW THAT AT THE Y-AXIS x=0
PLUG IN 0 IN THE PLACE OF X TO GET Y IN THE EQUATION OF THE LINE.
[tex]y = 2(0) + 4 \\ y = 4[/tex]
THE Y-INTERCEPT IS (0,4)
THEREFORE THE LINE HAS A SLOPE 2 AND A Y INTERCEPT (0,4)
FIRST OPTION IS THE ANSWER.
Gavin rides motocross in competition. A competition-level bicycle costs $1,800. He can borrow themoney from the bank at 3.596 interest for two years. By the end of the loan, how much money willGavin end up paying the bank?
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The simple interest formula is:
[tex]I=Prt\text{ }[/tex]where P is the principal, r is the interest rate and t is the time.
In this case the principal is $1800, the interest rate is 0.035 (in decimal form) and t is 2. then the interest he pays is:
[tex]I=(1800)(0.035)(2)=126[/tex]Therefore in total he will pay $1926
Hi! I am struggling on 41-44. Can you help me with 44?
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Let:
The coordinates for the new figure will be given by:
[tex]Do,k(x,y)=(k(x-a)+a,k(y-b)+b)[/tex]Where:
O = Center of dilation at (a,b) = (3,2)
k = Scale factor = 0.1
So:
[tex](1,1)->(0.1(1-3)+3,0.1(1-2)+2)=(2.8,1.9)[/tex][tex]\begin{gathered} (1,2)->(0.1(1-3)+3,0.1(2-2)+2)=(2.8,2) \\ (2,1)->(0.1(2-3)+3,0.1(1-2)+2)=(2.9,1.9) \\ (2,2)->(0.1(2-3)+3,0.1(2-2)+2)=(2.9,2) \end{gathered}[/tex]Since:
(2.8,2), (2.8,1.9), (2.9, 1.9) and (2.9,2) are much closer to the center of dilation we can conclude that the dilated figure is closer to the center of dilation.
Pls answer very desperate ty
The science club members are using transformations on coordinate grids to track the movement of constellations in the sky. Choose the left side of the constellation depicted by the line passing through (1,7) and (2.5,1), and find the linear functions that correspond to the series of motions of the side given below.
1) Shift the side down 5 units.
2) Vertically stretch the result of the shift by 3.
3) Shift the result of the stretch 2 units to the left.
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The linear function passing through points (1,7) and (2.5, 1) is given by:
y = -4x + 11.
What is a linear equation?
It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
y = mx + b
In which the coefficients are described as follows:
The coefficient m is the slope of the function, representing the rate of change of the function.The coefficient b is the y-intercept of the function, representing the value of y when the function crosses the y-axis(x = 0).For this problem, the two points that form the function are given as follows:
(1,7) and (2.5, 1).
The slope is given by change in y divided by change in x, hence it is given by:
m = (1 - 7)/(2.5 - 1) = -4.
Then:
y = -4x + b.
When x = 1, y = 7, hence we can find the intercept b of the function as follows:
7 = -4(1) + b
b = 7 + 4
b = 11.
Hence the equation is:
y = -4x + 11.
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A sea lion was swimming at 2 feet below sea level. The number line showsthe location of the sea lion. It then swam down 8 feet. Describe how to usethe number line to find the new location of the sea lion.109 8-7 654 3 -2-1 0 1 2 345 6789 10OA. On the number line, move 8 units to the right. End at 10. The sealion was 10 feet above sea level.OB. On the number line, move 8 units to the left. End at -6. The sealion was 6 feet below sea level.O C. On the number line, move 8 units to the left. End at -10. The sealion was 10 feet below sea level.OD. On the number line, move 8 units to the right. End at 6. The sealion was6 feet above sea level.EPREVIOUS
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Answer:
C is the correct statement. Since the sea lion started at 2 feet below sea level and went down another 8 feet, the sea lion is now at 10 feet below sea level.
The weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g.
a. For the bell-shaped graph of the normal distribution of weights of Hershey Kisses, what is the area under the curve?
b. What is the value of the median?
c. What is the value of the mode?
d. What is the value of the variance?
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a. The area under the curve is 1.
b. The value of the median is 4.5338.
c. The value of the mode is 4.5338.
d. The value of the variance is 0.0108.
How to illustrate the information?It should be noted that the weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g.
In this case, for a normal distribution, mean = median = mode. The median is 4.5338
The mode is also 4.5338.
The variance will be the square of the standard deviation. This will be:
= (0.1039)²
= 0.0108.
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8. Jen receives 8 dollars each week for allowance. She also had 35 dollars saved from birthday gifts. If she currently has 195 dollars, write an equation to represent how many weeks Jen has been saving money. a. 8w + 195 +35 b. 8 + 35 = 195w c. 8w + 35 = 195 d. 8w-35 = 195
Answers
the equation for the saved money is
195 = 8w + 35
here w = no. of weeks.
so the answer is 8w + 35 = 195
prove that 4^3 / 4^6 = 4^-3, without using the exponent law x^a / x^b = x^a-b.
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The second angle of a triangle measures three times as large as the first. If the third angle measures 55 degrees more than the first, find the measure of all three angles. ( recall that the sum of the angles of a triangle add to 180 degrees )
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Taking x as the measure of the first triangle, we know that the second one measures three times as x:
[tex]3x[/tex]And the third one measures 55 degrees more than the first:
[tex]x+55[/tex]We know that the sum of the angles of a triangle is 180, use this information to find x:
[tex]\begin{gathered} x+3x+(x+55)=180 \\ 5x+55=180 \\ 5x=180-55 \\ x=\frac{125}{5} \\ x=25 \end{gathered}[/tex]It means that the first angle measures 25°. Use this value to find the second and third angles:
[tex]\begin{gathered} 3x=3\cdot25=75 \\ x+55=25+55=80 \end{gathered}[/tex]It means that the second angle measures 75° and the third one measures 80°.
I NEED HELP ASAP!! i’m not sure how to do this.
Answers
Answer:
slope = -2/3
Step-by-step explanation:
Locations of points: (-3 , 2) and (3 , -2)
slope = rise / run
between two plotted points this means that:
slope = change in y / change in x
slope = (-2 - 2) / (3 - (-3))
slope = -4/6
reduce:
slope = -2/3
37 cm50°The measure of angle A is type your answer...29 cm
Answers
The Solution:
Given:
Required:
To find the measure of angle A.
By sine rule:
[tex]\frac{\sin A}{a}=\frac{\sin C}{c}[/tex][tex]\frac{\sin A}{37}=\frac{\sin50}{29}[/tex][tex]undefined[/tex]
L C R
U (4, 14) (9, 6) (5, 3)
M (8, 2) (6, 12) (1, 7)
D (11, 5) (16, 3) (9, 8)
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Explanation: huh
The area of a square is 256cm2What is the length of its side?
Answers
The area of a square is 256cm2
What is the length of its side?
Remember that
the area of a square is equal to
A=b^2
wheer
b is the length side
in this problem we have
A=256 cm2
substitute
256=b^2
square root both sides
b=16 cm
answer is
length side is 16 cmsolve this please ???
Answers
Answer:
I am a little confused because it says the point-slope formula, but then tells you to solve for y and to distribute. Attached , I put the answer in the point slope form, but if they want the equations solves for y, then the answer would be in the slope intercept form. Below the answers are in point slope form, but I will put the answer in slope intercept form here.
1 y = 2x -1
2. y = 1/2x +3
3. y = 2/3x + 4
4. y = -3/4x -7
5.y = -2x + 1
6. y = 1/3x -5
7. y = -5/3x + 7/3
8 y = 3x + 1
Step-by-step explanation:
Differentiate the equation to find the functions for
Velocity v= ds/dt
Answers
Differentiation from First Principles is a formal method for determining a tangent's gradient.
What is meant by differentiation?Finding a function's derivative is the process of differentiation. It is also the process of determining how quickly a function changes in relation to its variables.
According to the Sum rule, a sum of functions' derivatives equals the sum of those functions' derivatives. The derivative of two different functions is the difference of their derivatives, according to the Difference rule.
Differentiation from First Principles is a formal method for determining a tangent's gradient. The straight line connecting any two locations on the curve that are fairly near to one another will have a gradient that is similar to that of the tangent at those places.
s(t) = ∫vdt = ∫sin(πt)dt = (-cos(πt))/π + c
substituting the values t = 3, we get
s(-3) = (-cos(-3π))/π + c = 0
simplifying the above equation, we get
(-cos(3π))/π + c = 0
1/π + c = c
c = -1/π
Therefore, the correct answer is s(t) = (-cos(πt))/π - 1/π = (-cos(πt) - 1)/π.
The complete question is:
Given the velocity v=ds/dt and the initial position of a body moving along a coordinate line, find the body's position at time t. v=sin(pi*t), s(-3)=0
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I am thinking the answer is 8, because it's bisecting, and I was told when a bisection is taking place in an angle, the sides are congruent, and by just the look of the eye, they look the same length. If I'm wrong, could I get a explanation on how to find the length?
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BISECTRIZ: Divide the interior or exterior angle of each vertex into two congruent angles. Then we have that the angle that ES starts and that shares the 2 right triangles is the same:
As we can see, both triangles share the angle and the hypotenuse, that is, their height is also the same, so the height of PC is equal to 8
Looking at the graph, which point shows the constant of proportionality?
(2,1)
(2,4)
(,2)
(1,2)
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Answer:
I forgot maybe someone else knows
Step-by-step explanation:
:) ;) <*)))))<
WXYZ is a rectangle if M angle w x y equals 6X squared - 6 find a
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Given the rectangle WXYZ, the angle m∠WXY=6a²-6
The given angle is a corner angle, and as you might remember all corner angles of a rectangle are right angles, so we can say that the given expression equals 90 degrees:
[tex]6a^2-6=90[/tex]From this expression you can calculate the value of a.
First step is to add 6 to both sides of the equation so that the a-related term stays alone in the left side of the equation and all costants are in the other side:
[tex]\begin{gathered} 6a^2-6+6=90+6 \\ 6a^2=96 \end{gathered}[/tex]Next divide both sides by 6:
[tex]\begin{gathered} \frac{6a^2}{6}=\frac{96}{6} \\ a^2=16 \end{gathered}[/tex]And calculate the square to both sides of the variable to reach the possible value of a:
[tex]\begin{gathered} \sqrt[]{a^2}=\sqrt[]{16} \\ a=4 \end{gathered}[/tex]Now, just because the result is positiv, that does not mean that is the only possible value for a, if you square -4 you can also get 16 as a result, so a can be negative 4 or positive 4:
a=±4
The correct option is B.
Simplify the following expression. -7x²-2+5x+13x² - 15x
Answers
Answer: 2(3x^2-5x-1) or 6x^2-10x-2
Step-by-step explanation: combine like terms, and then factor by grouping :)
This math is new to me please help so I can take notes
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We have here two sets of numbers, and we have to find the value for one of the operations of sets, that is, the intersection operation.
The result of this operation is the values that are common to both sets.
If we saw set A and set B, we will have:
[tex]A=\mleft\lbrace1,4,8,13,15\mright\rbrace[/tex][tex]B=\mleft\lbrace2,3,5,12,17\mright\rbrace[/tex]Both sets have five elements. However, they have no common elements (or nothing in common). Therefore, the result will be the empty set:
[tex]A\cap B=\varnothing[/tex]That means both sets have no common elements ---> empty set.
In summary, we have that:
[tex]A\cap B=\varnothing[/tex]8.20y + 4) = 6.7s +5.2
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distribute the parentheses
[tex]\begin{gathered} 8.2y+32.8=6.7s+5.2 \\ \end{gathered}[/tex]substract 32.8 on both sides
[tex]undefined[/tex]Evaluate the expression (c^2) + (d^3) for c = 3 and d = 3
Answers
ANSWER
36
EXPLANATION
We want to evaluate the expression for when c = 3 and d = 3.
The expression given is:
[tex]c^2+d^3[/tex]To do this, we will simply replace the values of c and d with the values given:
[tex]\begin{gathered} 3^2+3^3 \\ =\text{ 9 + 27} \\ =\text{ 36} \end{gathered}[/tex]That is the answer.
Evaluate the following expressionsYour answer must be an exact angle in radians and in the interval Example: Enter pi/6 for lceil- pi 2 , pi 2 ] pi R
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determine the discriminat and use it to determine the number of x-intercepts for the graph of f(x) = 2x^2 - 8x + 9
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Answer:
[tex]\begin{gathered} x=2+i14 \\ x=2-i14 \\ \text{ Complex answer, no x-intercepts} \end{gathered}[/tex]Explanation: The discriminant is the part of the quadratic formula underneath the square root symbol As we know that square root of any number is both plus and minus of a certain value:
[tex]\sqrt[]{a}=\pm c\Rightarrow(-c)^2=(+c)^2=a[/tex]Using this information about the discriminant we can determine the x-intercepts of the graph as follows:
[tex]f(x)=2x^2-8x+9=0\Rightarrow\text{ Solving this quadratic equation will give x-intercepts}[/tex]Quadratic equation solution:
[tex]\begin{gathered} f(x)=2x^2-8x+9=0\Rightarrow x=\frac{-B\pm\sqrt[]{B^2-4AC}}{2A} \\ \because\Rightarrow \\ A=2 \\ B=-8 \\ C=9 \\ \therefore\Rightarrow \\ x=\frac{-(-8)\pm\sqrt[]{(-8)^2-(4\times2\times9)}}{2\times(2)}=\frac{8\pm\sqrt[]{16^{}-72}}{4}=\frac{8\pm\sqrt[]{-56}}{4}=2\pm\frac{\sqrt[]{-56}}{4} \\ \therefore\Rightarrow \\ x=2\pm i14 \\ \rightarrow\text{ Complex answer} \\ x=2+i14 \\ x=2-i14 \end{gathered}[/tex]Graph
Note! The plot of f(x) above confirms that there are no x-intercepts of this function, so the reason for complex values of x. also discriminant is the square root of -56
Can you help me solve this problem I know how to do the others but I can’t really figure this one out
Answers
We have an inequality, this inequality is as follows
[tex]4x\ge112[/tex]To solve the inequality we must clear "x" and see what it tells us
[tex]\begin{gathered} x\ge\frac{112}{4} \\ x\ge28 \end{gathered}[/tex]The inequality indicates that "x" belongs to the set of numbers equal to or greater than 28
Of the set of numbers we have, the only ones that meet this condition are numbers 28 and 33
In conclusion, Only options D and F are correct
The sum of six, and a number divided by two is 0
Answers
Answer:
let the unknown be x
=x+6/2=0
=0=x+6
=-x=6
divide both sides by -1
=-x/-1=6/-1
=x=-6
Joan invests $200. She earns interest at 3% per annum, compounded monthly.What is the future value of Joan’s investment after 1.5 years?
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For this exercise you need to use the following formula:
[tex]FV=PV\mleft(1+r\mright)^n[/tex]Where "FV" is the future value, "PV" is the present value, "r" is the interest rate (in decimal form), and "n" number of periods.
In this case, analyzing the information given in the exercise, you can identify that:
[tex]PV=200[/tex]Remember that a percent can be written in Decimal form by dividing it by 100. Then:
[tex]\frac{3}{100}=0.03[/tex]Since she earns interest at 3% per annum and it is compounded monthly, you can determine that the interest rate per month is:
[tex]r=\frac{0.03}{12}=0.0025[/tex]Knowing that 1 year has 12 months, you know that:
[tex](12)(1.5\text{ }years)=18\text{ }months[/tex]Then:
[tex]n=18[/tex]Therefore, you can substitute all those values into the formula and then evaluate, in order to find the future value of Joan’s investment after 1.5 years:
[tex]\begin{gathered} FV=PV(1+r)^n \\ FV=(200)(1+0.0025)^{18} \\ FV=(200)(1.0025)^{18} \end{gathered}[/tex][tex]FV\approx209.19[/tex]Hence, the answer is: $209.19 (approximately).
solve for x5x - 7 = 2x + 11
Answers
Answer:
Explanation:
Given the below equation;
[tex]5x-7=2x+11[/tex]To solve for x, the 1st step is to subtract 2x from both sides of the equation;
[tex]\begin{gathered} 5x-2x-7=2x-2x+11 \\ 5x-2x-7=11 \\ 3x-7=11 \end{gathered}[/tex]The 2nd step is to add 7 to both sides of the equation;
[tex]\begin{gathered} 3x-7+7=11+7 \\ 3x=18 \end{gathered}[/tex]The final step is to divide both sides of the equation by 3;
[tex]\begin{gathered} \frac{3x}{3}=\frac{18}{3} \\ x=6 \end{gathered}[/tex]Find the length, width, area, and aspect ratio of a SAMSUNG Flat 75-Inch 4K 8 Series UHD Smart TV with HDR and Alexa Compatibility.
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The dimensions that we must use in this problem, are the dimensions of the screen of a TV of 75 inches:
• W = width = 65.4" ,
,• H = heigh = 36.8".
• The form of the TV is a rectangle, we compute the area A of the rectangle by the following formula:
[tex]A=W\cdot H=65.4in\cdot36.8in=2406.72in^2.[/tex]• The aspect ratio r is given by the quotient of the width (W) by the height (H):
[tex]r=\frac{W}{H}=\frac{65.4in}{36.8in}=1.777.[/tex]• The length of the TV diagonal L can be computed using Pitagoras Theorem:
[tex]L=\sqrt[]{W^2+H^2}=\sqrt[]{(65.4in)^2+(36.8in)^2}\cong75.04in.[/tex]Answer
• width = 65.4",
,• heigh = 36.8",
,• area = 2406.72in²,
,• aspect ratio = 1.777
,• length of the diagonal = 75.04in.
Andrew is writing a coordinate proof to show that the triangle formed by connecting the midpoints of the sides of an isosceles triangle is itself an isosceles triangle. He starts by assigning coordinates as given.
Enter the answers in the boxes to complete the coordinate proof.
P is the midpoint of DE. Therefore, the coordinates of P are ( answer, b ).
Q is the midpoint of DF. Therefore, the coordinates of Q are (3a, answer).
R is the midpoint of EF. Therefore, the coordinates or R are ( answer, answer)
The length of PR is √ a^2+b^2. The length of QR is √ a^2 +b^2.
Comparing the expressions for the lengths of PR and QR shows that the lengths are equal. therefore, △PQR is isosceles
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The area of triangle DEF = 4 ( area of triangle QRP) which is determined that by comparing the expression for the lengths of PR and QR.
What are congruent triangles?Two triangles are said to be congruent if their corresponding sides and angles are equal.
According to the data provided,
Triangle ΔDEF is an isosceles triangle having DE, EF, and FD sides.
Since DE = EF and P,R, and Q are the midpoints of the sides DF, EF, and DE,
We know from isosceles triangle theorems that the area of a triangle formed by uniting the midpoints of two isosceles triangles is one-fourth the area of the isosceles triangle.
As evidence, consider the following:
Because point Q is the halfway of DE, its coordinates are (a, b). (Because the midpoint of a line segment always has coordinates that are half the total of the coordinates of the end points.)
Because point R is the midpoint of FE, its coordinates are (3a,b).
The length of the base, DF, in triangle DEF is 4a and the height is 2b. As a result, its area is 4ab. (Because the area of a triangle equals 1/2 its base × height.)
The length of the base, QR, of the triangle QRP is 2a, while the height is b.
As a result, its area is ab.
Therefore, the area of triangle DEF is four times that of triangle QRP.
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