PLZ HELP, DUE TONIGHT!! Find the values of a and b such that f(x) is continuous at x=
Answers
Answers:
a = 2 and b = -4
============================================================
Explanation:
Let's define the three helper functions
f(x) = ax^2 - bh(x) = 6j(x) = 5ax+bwhich are drawn from the piecewise function. The g(x) function will change depending on what the input is.
If x < 1, then g(x) = f(x). If x = 1, then g(x) = h(x)If x > 1, then g(x) = j(x)Since we want g(x) to be continuous at x = 1, this must mean the three functions f(x), h(x), j(x) must have the same output value when the input is x = 1.
Because h(x) = 6 is a constant function, the output is always 6 regardless of the input. Therefore, we want f(x) and j(x) to have 6 as their output when x = 1. Or else, the pieces won't connect.
Plug x = 1 into the f(x) function to get
f(x) = ax^2 - b
f(1) = a(1)^2 - b
f(1) = a - b
Set this equal to the desired output of 6 and we end up with the equation a-b = 6. Solving for 'a' leads to a = b+6.
------------
We'll use the same idea for j(x)
j(x) = 5ax + b
j(1) = 5a(1) + b
j(1) = 5a + b
5a+b = 6
5(b+6) + b = 6 ... plug in a = b+6; solve for b
5b+30+b = 6
6b+30 = 6
6b = 6-30
6b = -24
b = -24/6
b = -4
Which then leads to,
a = b+6
a = -4+6
a = 2
------------
Since a = 2 and b = -4, we go from this
[tex]g(x) = \begin{cases}ax^2-b, \ \ x < 1\\6, \ \ x = 1\\5ax+b, \ \ x > 1\end{cases}[/tex]
to this
[tex]g(x) = \begin{cases}2x^2+4, \ \ x < 1\\6, \ \ x = 1\\10x-4, \ \ x > 1\end{cases}[/tex]
Meaning
f(x) = 2x^2+4 and j(x) = 10x-4
You should find that plugging x = 1 into each of those two functions leads to 6 as the output.
The graph is shown below. Note the red graph f(x) is only drawn when x < 1. Similarly, j(x) is only drawn when x > 1. The orange point represents h(x) which only happens when x = 1. So as the name implies, the piecewise function g(x) is composed of pieces of the three functions f(x), h(x), j(x).
Related Questions
According to a recipe for purple paint, the ratio of milliliters of blue paint to milliliters of red paint in the mixture should be 20:4. Selena has 140 ml of blue paint. She wants to mix red paint to it to get the same shade as suggested by the recipe.
For every 1 ml of red paint, the recipe suggests adding [Drop Down 1] of blue paint. So, for
140 ml of blue paint the amount of red paint needed is [Drop Down 2].
Answers
Answer:
30ml
Step-by-step explanation:
A new 54 inch flatscreen TV cost $1245 at Best Buy. If Harry has a 15% off coupon what will he pay for the TV
Answers
Answer: 1,058
Step-by-step explanation:
Evaluate (2x + 3) + ( x- 5) show answer pls
Answers
Step-by-step explanation:
2x +3 +x-5
3x+3+(-5)
3x +(-2)
x= -2÷3
x=-6
HOPE IT HELPEDHelp pleaseee anyone??
Answers
It 's c my friend
okayyyyyyyyyyyyyyy
The dot plot shows how many cats are owned by each person in a neighborhood: Dot plot titled Neighborhood Cats and labeled Number of Cats shows number line from 1 to 10. There are 6 dots over 1, 3 dots over 2, 2 dots over 3, 1 dot over 4, 0 dots over 5, 6, 7, 8, and 9, and 1 dot over 10. Is the median or the mean a better measure of center for these data and why
Answers
Answer:
Median, because the data are skewed and there is an outlier
Step-by-step explanation:
yo hoped it helped
One batch of walnut muffins use 1 1/3 cups of walnuts.How many cups of walnuts are needed to 3 3/4 batches of muffins, answers are only 5 cups, 3 3/12 cups, 5 1/12 cups, and 4 cups, there are no 3 1/4 cups
Answers
Answer:
5 cups
Step-by-step explanation:
Let us set it up as a fraction.
[tex]\frac{\frac{11}{3}}{1}=3\frac{\frac{3}{4}}{x}[/tex]
Step 1: Divide the given values
[tex]\frac{11}{3}\div 1 \\\\=\frac{11}{3}[/tex]
Step 2: Multiply the result by the other value
[tex]3\frac{3}{4} \\\\3\frac{3}{4} \times \frac{11}{3}\\\\= 13\frac{1}{4}[/tex]
So, with this solution, the answer is [tex]13\frac{1}{4}[/tex]. However, the answer isn't included in the options provided, so that means the answer is 5 cups.
Mi papá quiere comprar una camioneta que cuesta $373,990. Tiene que dar de enganche el 28%. ¿Cuánto es lo que tiene que pagar?
Answers
Answer:
i don't speak spanish sorry
Step-by-step explanation:
Answer:
104,717
Step-by-step explanation:
multiplica 373990 por 28% es .28
I need help
An amount of $16,000 is borrowed for 8 years at 7.25% interest, compounded annually. If the loan is paid in full at the end of that period, how much must be paid back?
Answers
Answer:
$28009.05
Step-by-step explanation:
Let us use the formula A = P(1 + r/n)^(nt)
Step 1: Convert 7.25 to decimal
[tex]7.25\% = 0.0725[/tex] in decimal form
Step 1: Percent means 'per 100'. So, 7.25% means 7.25 per 100 or simply 7.25 over 100. [tex]7.25 \div 100\\\\= \frac{7.25}{100}\\\\= 0.0725[/tex] If you divide 7.25 by 100, you'll get 0.0725 (a decimal number).Step 2: Solve with the formula
[tex]A = 16000(1 + 0.0725)^8\\\\A= 28009.05[/tex]
Step 1: Add the numbers: [tex]1+0.0725=1.0725[/tex][tex]=16000\cdot \:1.0725^8[/tex][tex]=16000\cdot \:1.75056[/tex]Step2: Multiply[tex]=28009.05068[/tex]A television shop sold a set for $799. It made a profit of $ 245.25. How much did the shop buy the television for ?
Answers
Answer:
the shop originally bought the television for $553.75
Step-by-step explanation:
this is because if they sold it for $799 and made a profit of $245.25 then you would subtract the profit from the price the TV sold for and you get $553.75
Please help! I will give brainliest to the first correct answer!
Answers
Answer:
A)
Step-by-step explanation:
check that the formula in A works . . .
the 7th term should be 320
a(320) = 5(2)^7-1
= 5(2)^6
= 5(64)
= 320
HELP DUE IN 15 MINS!
Write the equation of a circle that has a center of (-3, 4) and goes through the point (2, 1).
Answers
Answer:
Step-by-step explanation:
You can find the left side of the equation for a circle easily enough. It's r^2 that's the problem. It is on the right.
Left Side
(x + 3)^2 + (y - 4)^2 = r^2
Right Side.
x1 = - 3
x2 = 2
y1 = 4
y2 = 1
Use the distance formula to get r^2
r^2 = (x2 - x1)^2 + (y2 - y1)^2
r^2 = (2 - - 3)^2 + (1 - 4)^2
r^2 = 5^2 + (-3)^2
r^2 = 25 + 9
r^2 = 34
Answer
(x + 3)^2 + (y - 4)^2 = 34
Select ALL of the expressions that are equivalent to 3(8-4x) A. (4x+ 8) + (4x+8) +( 4x+8) B. 24+12x C. -4x+24 D. 12x+24 E. - 12x-24
Answers
A. Should be 8 - 4x, not 8 + 4x
B. 3(8 - 4x) = 24 - 12x not 24 + 12x
C. Doesn’t make sense at all
D. 3(8 - 4x) = 24 - 12x Correct!
E. 3(8 - 4x) = 24 - 12x 24 shouldn’t be negative
There are 3 oranges, 8 peaches, and 4 apples in a fruit basket. David selected one piece of fruit at random. He was hungry and ate that fruit. He then selected a second piece of fruit. What is the probability that he selects two peaches?
Answers
Answer:
Probabilty[First and second fruit is peach without replacement] = 0.267 (Approx.)
Step-by-step explanation:
Given:
Number of oranges = 3
Number of peach = 8
Number of apple = 4
Find:
Probabilty[First and second fruit is peach without replacement]
Computation:
Probablity of an event = Number of favourable outcomes / Number of outcomes
Probabilty[First fruit is peach] = 8 / [3 + 8 + 4]
Probabilty[First fruit is peach] = 8 / 15
Probabilty[Second fruit is peach] = [8-1] / [3 + 8 + 4 - 1]
Probabilty[Second fruit is peach] = 7 / 14
Probabilty[Second fruit is peach] = 1 / 2
Probabilty[First and second fruit is peach without replacement] = Probabilty[First fruit is peach] x Probabilty[Second fruit is peach]
Probabilty[First and second fruit is peach without replacement] = [8/15][1/2]
Probabilty[First and second fruit is peach without replacement] = 8 / 30
Probabilty[First and second fruit is peach without replacement] = 0.267 (Approx.)
does anyone understand this?
Answers
Answer:
Hello! answer: 80
Step-by-step explanation:
When doing the Pythagorean theorem you do...
a^2+b^2=c^2 which is just A multiplied byitself + B multiplied byitself that will equal the number that C multiplied by itself would equal then you just figure out what multiplied byitself equals that number and thats the answer so...
64 × 64 = 4096 48 × 48 = 2304
4096 + 2304 = 6400 80 × 80 = 6400 therefore c = 80 hope that helps!
Answer:
c = 80 inches
Step-by-step explanation:
The length of the hypotenuse can be found by using the following formula in which "c" is the length of the hypotenuse, and "a" and "b" are the legs of the right angles triangle.
[tex]c = \sqrt{a^{2} + b^{2} }[/tex]
[tex]c= \sqrt{64^{2} + 48^{2} } \\c= \sqrt{4096 + 2304} \\c = \sqrt{6400} \\c = 80 inches[/tex]
Find the value of x.
Answers
Answer:
70.86 degrees
Step-by-step explanation:
cos x = adjacent side / hypotenuse
= 2/6.1 = 0.32787
x = 70.86.
Answer:
x ≈ 70.9°
Step-by-step explanation:
Using the cosine ratio in the right triangle
cos x = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{2}{6.1}[/tex] , then
x = [tex]cos^{-1}[/tex] ([tex]\frac{2}{6.1}[/tex] ) ≈ 70.9° ( to 1 dec. place )
What is the reciprocal function of sine?
A. Cosine
B. Cotangent
C. Cosecant
D. Secant
Answers
Answer: its a C
Step-by-step explanation: The cosecant is the reciprocal of the sine. It is the ratio of the hypotenuse to the side opposite a given angle in a right triangle.
help plzzzz i need to find the volume of the cone
Answers
Answer:
2h3-r5
Step-by-step explanation:
first 3-2=5
then 2h3-r=2h3-r5
finally 2h3-r5
Answer:
3.28653¯V7h
Step-by-step explanation:
Paul ran 12 miles in 1 hour and 36 minutes. What was Paul's average rate, in minutes per mile?
Answers
Answer:
8 miles per minute
hope this helps
have a good day :)
Step-by-step explanation:
Answer:
8 minutes per mile
Need help thanks please :)
Answers
Answer:
t:
[tex] \infty , - \infty [/tex]
meaning any number can substitute t and it would be the same answer.
Here’s the revenue and expenses for the month. Calculate whether Mia had a profit or loss
Answers
Answer:
300
Step-by-step explanation:
subract costs from revenue
Can someone please help me?
What is the length of the side
between point A and point B?
I
15 ft A
18 ft
B.
9 ft
27 ft
Answers
Answer:
195 answer.plus minus plus minus plus minus
If a function is defined by the equation y=-5x+1 and the domain is the set {-2,0,2,4},
which set below represents the range of the function?
(1) {11, 12, 13, 14}
(3) {-10, -1,9,19)
(2) {-19, -9,1,10)
(4) {51, 53, 55, 57)
Answers
A
Identify the input values.
Since there is an even root, exclude any real numbers that result in a negative number in the radicand
The solution(s) are the domain of the function.
Mark Me Brainest
What is the perimeter of the house is c=5
I’m marking brainliest.
Answers
Answer:
132
Step-by-step explanation:
C=5
3c - 2 = 13
2c = 10
4c + 1 = 21
4c + 2 = 22
length = (3c- 2) + (4c + 1)
length = 13 + 21
length = 34
width = (4c + 2) + (2c)
width = 22 + 10
width = 32
Perimeter
p = 2l + 2w
p = 2*34 + 2*32
p = 68 + 64
p = 132
What is the length of the missing leg?
Answers
Answer:
a = 6
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityTrigonometry
[Right Triangles Only] Pythagorean Theorem: a² + b² = c²
a is a leg b is another leg c is the hypotenuseStep-by-step explanation:
Step 1: Define
Identify variables
a = a
b = 8
c = 10
Step 2: Solve for a
Substitute in variables [Pythagorean Theorem]: a² + 8² = 10²Evaluate exponents: a² + 64 = 100[Subtraction Property of Equality] Subtract 64 on both sides: a² = 36[Equality Property] Square root both sides: a = 6Please solve this for me!
4c + 36 divided by 3 + 4(1 + 2) if c = 5
Answers
Answer:4 or 3.70
Step-by-step explanation:
well if c = 5 you should 1st do 4 x 5= 20 + 36 and that equals 56
now do (1+2)=3 x 4 =12 + 3 =15
now not sure what ur answers are but i got either 4 or 3.70
Good Luck!!!
R of breaths of all people in asia in 2016? Expressing your answer in scinentific notation in the form a *10^(b) what are the values of a and b?
Answers
Answer:
[tex]a = 3.7464[/tex]
[tex]b = 16[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required
Express the answer in [tex]a * 10^b[/tex]
From the question, we have:
[tex]People = 4.46 * 10^9[/tex]
[tex]Breath = 8.4 * 10^6[/tex] breath/person in a year
The number of breath is calculated by:
[tex]Overall = People * Breath[/tex]
So, we have:
[tex]Overall = 4.46 * 10^9 * 8.4 * 10^6[/tex]
Reorder
[tex]Overall = 4.46 * 8.4* 10^9 * 10^6[/tex]
[tex]Overall = 37.464* 10^{9+6}[/tex]
[tex]Overall = 37.464* 10^{15}[/tex]
Using scientific notation [tex]a * 10^b[/tex];
The value (a) must be in the following interval [tex]0 \le a \le 1[/tex]
So, we have:
[tex]Overall = 3.7464 * 10 * 10^{15[/tex]
[tex]Overall = 3.7464 * 10^{16[/tex]
By comparing the above with: [tex]a * 10^b[/tex]
[tex]a = 3.7464[/tex]
[tex]b = 16[/tex]
Santiago buys a new television for $989.76. If
the sales tax rate is 8.75%, what is the total cost
of the television?
Answers
Answer:
$1076.36
Step-by-step explanation:
The full price before tax can be represented by 1.00($989.76), and the 8.75% tax by 0.0875($989.76)
which can be combined to obtain the total cost: (1.0875)($989.76) = $1076.36
1/2 divided by 4 =__
Answers
1/2 dived by 4 = 0.125
Answer:
1/2 divided by 4 = 1/8 or 0.125
Step-by-step explanation:
1/2 divided by 4
(1/2) ÷ 4
= 1/2 × 1/4
= 1/8 or 0.125
Jack can program an app in 3 hours. Chris can program the same app in 8 hours. How long would it take them to program the same app working together?
Answers
Answer:
2 2/11 hours
Step-by-step explanation:
Let the number of hours
Jack spent programming his app = J
Chris spent programming his app = C
Total hours if they worked together = T
Hence,
1/T = 1/3 + 1/8
Lowest Common Denominator = 34
1/T = 8 + 3/24
1/T = 11/24
Cross Multiply
T × 11 = 1 × 24
T = 24/11
T = 2 2/11 hours
Therefore, it would take them to program 2 2/11 hours to program the same app working together
Lynn spent $21.90 at A Hill of Beans. She bought 2 fruit smoothies at a
cost of $8.90. She bought 2 bear claws at $1.75 each. The rest of the
money was spent on 2 iced mochas. What is the cost of one iced
mocha?
Answers
Answer:
$0.3
Step-by-step explanation:
8.90x2=17.8
1.75x2=3.5
17.8+3.5=21.3
21.90-21.3=0.6
0.6/2=0.3