Mathematics College

Please thank you :(Use the functions f(x) = x ^ 2 - 1 and g(x) = x + 11 to answer parts (a) - (g)

Answers

Answer 1

Answer:

(a) {-1, 1)

(b) {-11}

***(d) { x < -1 or x > 1, I am not sure how else to put it, try {-1, 1} if you are not allowed to put in > or < signs

(e) (-∞, -11] or x ≤ -11

(f) {-3, 4}

(g) ( -√2, √2)

***I know the correct solution sets but I am not sure how to represent all of them given the one box you have in your solution for (a). Maybe the solution boxes for the other parts are different. Without knowing that I am only able to provide what is the correct expression

Step-by-step explanation:

We have
f(x) = x² - 1

g(x) = x + 11

(a) f(x) = 0

=> x² - 1 =

=> x² = 1

x = ± 1

Answer (a): The solution set is {-1, 1}

(b)
g(x) = 0

=> x + 11 = 0

=> x = -11

Answer (b): Solution set: {-11}

This is a quadratic equation which can be solved by factoring:
=> (x - 4)(x+3) = 0

=> x = 4, x = -3

{-3, 4} Answer

(d) f(x) > 0
=> x² - 1 > 0
Add 1 to both sides

x² > 1

x < -1 or x > 1

[tex]\left(-\infty \:,\:-1\right)\cup \left(1,\:\infty \:\right)[/tex]

(e) g(x) ≤ 0
=> x + 11 ≤ 0

=> x ≤ -11

{-∞, -11}

(f)f(x) > g(x)

=> x²-1 > x +11

=> x² -x - 12 > 0

=> (x + 3)(x +4) > 0

=> x + 3 >0 and x - 4 > 0

=> x > -3 and x < 4

{-3, 4}

(g) f(x) ≥ 1

=> x² - 1 ≥ 1

=> x² ≥ 2

=> x ≥ ± √2

=> x ≤ -√2 or x ≥ √2

=> { -√2, √2}

In the above derivations note that if x² ≥ 1 then two possibilities are
x < - 1 or x > 1

Related Questions

I want the answer for this question pls...

Simplify 72h ÷ 9h

Answers

Answer:

Step-by-step explanation:

[tex] \frac{72h}{9h} = 8[/tex]

The h is common in the numerator and the denominator, so it cancels out. We are left with 72/9 = 8.

Juan is a teacher who has an annual salary of $65,798. He has to pay state income tax of 5%. What is the amount that Juan pays in state income tax?
Responses

Answers

Answer:

$3,289.90

Step-by-step explanation:

65,798 times 5%

How can you use the Product of Powers, Power of a Product, and Power of a Power Laws to evaluate and write equivalent numerical expressions with whole-number exponents?

Answers

The product of powers, power of a product, and power of a power laws can be used to evaluate the equivalent numerical expressions such as:

2² × 2³ = 2^5 = using product of Powers2² × 3² = (2×3)² = 6² = using power of a Product(2²)³ = 2^6 = using power of a power

How to illustrate the information?

According to the Power of a Product Property, you can distribute the exponent to each factor if a term is raised to one.

According to the rule of a power, when a base is raised to a power and then raised to another power, the exponents are multiplied while the base is left unchanged.

According to the product of powers rule, you can multiply two words with the same base by simply adding their exponents to get the result.

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The point (-13, 24) is on the graph of y = f(x). Find the corresponding point on the graph of y= g(x), where g(x) = 1/4f(x)

Answers

Answer:

(−12,2)

Step-by-step explanation:

Dividing the function by 2 divides all the y-values by 2 as well. So to get the new point, we will take the y-value

(4) and divide it by 2 to get 2

Therefore, the new point is

(−12,2)

Subtracting 2 from the input of the function makes all of the x-values increase by 2 (in order to compensate for the subtraction). We will need to add 2 to the x-value

(−12) to get −10

Therefore, the new point is

(-10,4)

Making the input of the function negative will multiply every x-value by

−

. To get the new point, we will take the x-value

(−12) and multiply it by −1 to get 12.

Therefore, the new point is

(12,4)

Multiplying the input of the function by 4 makes all of the x-values be divided by 4 (in order to compensate for the multiplication). We will need to divide the x-value

(−12) by 4 to get −3

Therefore, the new point is

(−3,4)

Multiplying the whole function by

increases all y-values by a factor of

, so the new y-value will be

times the original value

(4), or 16.

Therefore, the new point is

(−12,16)

Multiplying the whole function by

−1

also multiplies every y-value by

−1

, so the new y-value will be

−1

times the original value

(4), or −4.

Therefore, the new point is

(−12,−4)

Write the numbers in the blanks that make this equation true.

( 2 + 1 ) + 3 = 2 + ( _ + _ )

Answers

(2+1)+3=2+(2+2)

Any two numbers that add to 4.

Can someone explain how I would get the answer to this?

Answers

We are given the following values of the function to be

[tex]\begin{gathered} f(x)=7x^2-2x+5 \\ g(x)=-x+2 \end{gathered}[/tex]

To find (f.g)(x)

We will find the product of f(x) and g(x)

[tex]f(x)\times g(x)=(7x^2-2x+5)\times(-x+2)_{}[/tex]

Upon expanding, we will have

[tex]\begin{gathered} =7x^2\mleft(-x\mright)+7x^2\cdot\: 2-2x\mleft(-x\mright)-2x\cdot\: 2+5\mleft(-x\mright)+5\cdot\: 2 \\ =-7x^3+16x^2-9x+10 \end{gathered}[/tex]

Thus, the answer is:

[tex]=-7x^3+16x^2-9x+10[/tex]

Need help with this !!!!

Answers

The equation of the line in slope intercept form is y = -7x/2 - 5.

What is termed as the slope intercept form?A straight line's slope-intercept form is y = mx + b. In the equation y = mx + b for a straight line, m is known as the slope and b is known as the y-intercept. where y = mx+by = how much further up or down the line is,x = how far along the line is it,b = value of y when x equals 0 andm = indicates how steep this same line is.This is calculated using m = (distinction in y coordinates)/ (difference in x coordinates).

For the given question,

The graph of the straight line is given.

Select two points which lies on the line.

(x₁, y₁) = (0, -5)

(x₂, y₂) = (-2, 2)

Find the slope using the formula.

m = (y₂ - y₁)/(x₂ - x₁)

m = (2 + 5)/(-2 - 0)

m = -7/2

The equation of the line in slope intercept form is given as;

y - y₁ = m(x - x₁)

Put the values.

y + 5 = (-7/2)(x - 0)

y = -7x/2 - 5

Thus the equation of the line in slope intercept form is y = -7x/2 - 5.

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I have another question for brainly users out there.
Determine the value of y in the inequality.

12 + 4y < 32
Will mark brainliest.

Answers

Answer:

y = 3x-8

Step-by-step explanation:

A’ (,)

B, (,)

C’ (,)

Please answer I give brainliest

Answers

The image's coordinates are:

A = (-2,5),

B = (3,-3),

C = (-4,-1).

What is the Cartesian coordinate system?

A Cartesian coordinate system in a plane is a system of coordinates that uniquely identifies each point by a pair od numerical coordinates, that are the signed distances from two lines that are fixedly perpendicular to the point and are measured in the same length.

How do you find coordinates on a plane?

The reverse is done in order to determine a point's coordinates in a coordinate system. Start at the point and move along a vertical line either up or down to the x-axis. Your x-coordinate is located there. To get the y-coordinate, repeat the previous steps while following a horizontal line.

According to the given data:

The graph's coordinates are as follows:

A = (-2,5)

B = (3, -3)

C = (-4,-1)

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Find the distance between (-12, 1) and (12, -1).

Answers

If the coordinates of two points are (x_1,y_1) and (x_2,y_2), the distance between them is given by the formula:

[tex]d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Substitute for the coordinates: (-12,1) and (12,-1):

[tex]\begin{gathered} d=\sqrt[]{(12-(-12))^2+(-1-1)^2} \\ =\sqrt[]{(12+12)^2+(-2)^2} \\ =\sqrt[]{(24)^2+4} \\ =\sqrt[]{576+4} \\ =\sqrt[]{580} \\ =2\cdot\sqrt[]{145} \\ \approx24.08318916\ldots \end{gathered}[/tex]

Therefore, the distance between (-12,1) and (12,-1) is approximately 24.08.

Lynette is making candles for her online store the wholesale cost of the materials for one candle is $5.35 if she sells a candle for $12 what is the percent markup round to the nearest percent

Answers

The percent markup is 124.30%.

The wholesale cost of one candle is $5.35

The selling price of the candle is $12

The profit she earns = 12 - 5.35

= $6.65

Here we have to find the percent markup.

The formula for the percent markup =( profit/ wholesale cost )× 100

Markup percent = 6.65 / 5.35 × 100

= 1.24299 × 100

= 124. 30%

Therefore the markup percentage is 124.30%.

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A rectangular dog pen is constructed using a barn wall as one side and 120m of fencing for the other three sides. a) give a sketch of the situation. b) express the area in terms of x. c) sketch the graph. d) find the value of x that gives the greatest area. e) find the dimensions of the pen. f) state the domain.

Answers

Answer:

see the attachments for the sketch and the graphA = x(120 -2x)x = 30 maximizes the areadimensions: 30 m by 60 mdomain of x: [0, 60]

Step-by-step explanation:

You want an expression for area, maximum area, and dimensions of a rectangular pen fenced on three sides with a total of 120 m of fencing.

a) Sketch

The first attachment shows a sketch of the pen with x defined as the dimension the pen extends from the barn wall. The total length of fence is 120 m, so the side parallel to the barn will be (120 -2x) m in length.

b) Area

The area is the product of the length and width of the rectangle:

A = (x)(120 -2x)

c) Graph

The second attachment shows a graph of the area as a function of x.

d) Greatest area

The value of x that gives the greatest area is the x-coordinate of the vertex of the parabola. It is halfway between the zeros at x=0 and x=60. The maximum area will be had when x=30.

e) Dimensions

The dimensions of the pen are ...

x = 30

120 -2x = 120 -2(30) = 60

The pen is 30 m by 60 m, with the 60 m dimension parallel to the barn wall.

f) Domain

The area function only makes sense for values of x between 0 and 60 meters.

The domain of x is 0 ≤ x ≤ 60.

The following is the prime factorization of which composite number?
22.3².5
120
60
20
180

Answers

The prime factorization given as 2².3².5 is for the number D. 180.

What are prime numbers?

It should be noted that prime numbers are numbers that can only be divided by itself and one. This include 2, 3, 5, 7, etc

A composite number is one that can be generated by multiplying two smaller positive numbers. It is a positive integer with at least one divisor other than 1 and itself. Prime factorization is defined as the process of determining a number's prime factors so that the original number is evenly divisible by these factors.

In this case, it should be noted that we are given the expression as 2². 3². 5. It simply means that we have to multiply them to get the original number.

This will be:

= 2² × 3² × 5

= 4 × 9 × 5

= 180

Therefore, the number is 180.

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What is the slope of the line perpendicular to the line represented by the equation 2x+2y=-12

Answers

The slope of the line perpendicular to the line given by equation 2x+2y=-12 is found as 1.

What is the definition of line slope?The slope of a line measures its steepness as well as direction.The slope of a line is defined as the change throughout y coordinate in relation to the transitions in x coordinate.The net change in y coordinates is Δy, while the net change in x coordinates is Δx.So m represents the change in y coordinate in relation to a shift in x coordinate.

The slope can be calculated using the following formula:

m = Δy/Δx

Where, m is the slope.

The equation of the line is; 2x+2y=-12

Write the equation in slope-intercept form.

2x + 2y = -12

Divide by 2.

x + y = -6

y = -x - 6

Slope m₁ = -1 and y-intercept = -6.

As, both line are perpendicular.

Thus,

m₁ x m₂ = -1

m₂ = -1/m₁

Put the value.

m₂ = -1/(-1)

m₂ = 1

The slope of the perpendicular line is found as 1.

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The price of an item has been reduced by 60%. The original price was $65. What is the price of the item now?

Answers

Answer:

$26

Step-by-step explanation:

65 x 60% = 39

65 - 39 = 26

A square has a perimeter of 120 cm what is the length of one side of the square

Answers

The measure of each side of the square is 30cm

Carlos has a smart phone data plan that costs $20 per month that includes 7 GB of data, but will charge an extra $15 per GB over the included amount. How much would Carlos have to pay in a month where he used 4 GB over the limit? How much would Carlos have to pay in a month where he used went over by x GB?

Answers

Answer:

Carlos will end up paying 80£ in that month, the standard data plan cost 20£ plus 60£ the price of the extra 4 GB of data used.

Carlos has to pay total $80 in a month if he uses 4 GB data over the limit and if he uses x GB data over the limit then have to pay $20+15x.

What is ratio?

Ratio basically compares quantities, that means it show value of one quantity with respect to other quantity.

If a and b are two values, their ratio will be a:b,

Given that,

Cost of 7 GB data plan = $20,

Cost for extra data = $15 per GB.

Since, Carlos use extra 4 GB data in a month,

Cost of 1 GB extra data = $15

Cost of 4 GB extra data = $60

Total cost = cost of 7 GB data plan + Cost of 4 GB extra data

= $20 + $60

= $80

Total $80 Carlos have to pay, where he used 4 GB over the limit.

If Carlos uses x GB extra data,

Then cost of x GB extra data = $15x

Total cost = cost of 7 GB data plan + Cost of x GB extra data

= 20 + 15x

Total $20+15x Carlos have to pay, where he used x GB over the limit.

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Which of these strategies would eliminate a variable in the system of equations?
2x + 3y = -5
2x - 3y = 10
Choose all answers that apply:

Answers

Answer:

jhhuibhhhiihhuuuuuuuuuuuuuuuuuuuuuuuu

a) What is the difference between a 90º counterclockwise rotation and a 90º clockwise rotation?

b) If you use the letter C, what letter will each of those rotations look like...
...a 90º counterclockwise rotation?
...a 90º clockwise rotation?

Answers

Answer: answer to A: Clockwise rotations clockwise, while counterclockwise rotations rotate in the opposite direction

Answer to B: i have no idea but i hope the first one helps

Step-by-step explanation:

Shelby, Allen, and Denise printed their digital photos each requested the same size prints. Allen paid $3.24 for 11 prints. Denise paid $4.05 for 20 prints describe how you would write the equation for the relationship between the number of photos ordered and the amount paid

Answers

Taking x and y as the price for each photo, then the equation that represents each situation would be:

For Allen

[tex]11x=3.24[/tex]

For Denise

[tex]20y=4.05[/tex]

Three times the first of two numbers decreased by the second number equals 9. Twice the first number increased by the second number equals 11. Find the two numbers

Answers

As per the given condition for the system of equation, the two numbers are equal to 4 and 3.

As given in the question,

To begin with the solution, let the two unknown numbers be x and y. Now, translating the two mathematical statements into system of equations,

When three times the first of two numbers decreased by the second number equals 9.

3 x - y = 9 ____(1)

Twice the first number increased by the second number equals 11,

2 x + y = 11 ____(2)

By solving resultant system of equations we get,

3 x - y = 9

2 x + y = 11

5x = 20 (by adding the two equations)

=> x = 4

Substituting the value of x in second equation,

y = 11 - 2x

= 11 - 2(4)

= 11 - 8 = 3

Therefore, as per given condition for the system of equation, the two numbers are equal to 4 and 3.

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The Royal Fruit Company produces two types of fruit drinks. The first type is 40% pure fruit juice, and the second type is 65% pure fruit juice. The company is attempting to produce a fruit drink that contains 70% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 70 pints of a mixture that is 55% pure fruit juice?

Answers

By using the concept of weighted averages, 44.667 pints of 40 % pure fruit juice and 25.333 pints of 70 % pure fruit juice are needed to produce 70 pints of 55 % pure fruit juice.

How to produce a quantity of a given concentration from two quantities of different concentrations?

The situation describes a mixing process with two fruit juices of different concentrations. All pure fruit percentages are known and we are asked to determine the numbers of pints of 40 % pure fruit juice and 65 % pure fruit juice to produce 70 pints of 55 % pure fruit juice. This can be determined by means of the concept of weighted average:

(55 / 100) · 70 = (40 / 100) · x + (65 / 100) · (70 - x)

38.5 = 0.4 · x + 45.5 - 0.65 · x

0.15 · x = 7

x = 46.667

We need 44.667 pints of 40 % pure fruit juice and 25.333 pints of 70 % pure fruit juice to produce 70 pints of 55 % pure fruit juice.

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The amount of a radioactive isotope present at time t is given by A(t) = 500e^ -0.028171
grams.
How many grams remain after 40 years?

Answers

According to the radioactive isotope disintegration model, the remaining mass of the radioactive isotope after 40 years is approximately equal is 129.622 grams.

What is the amount of the mass of a radioactive isotope after a period of 40 years?

The deceasing exponential expression described in the statement models the mass of the radioactive isotope (m), in grams, as a function of time (t), in years, which represents the solution of an ordinary differential equation that represents a simple disintegration.

Now, we need to evaluate the function at t = 40 to find the amount remaining after 40 years:

A(40) = 500 · exp(- 0.028171 · 40)

A(40) = 129.622

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A school is arranging chairs in rows for an assembly. $11$ chairs make a complete row, and right now there are $110$ chairs total. The school wants to have as few empty seats as possible, but all rows of chairs must be complete. If $70$ students will attend the assembly, how many chairs should be removed?

Answers

Answer:

Step-by-step explanation:

Ela enjoys scuba diving. On one particular trip she was diving at a rate of 5 ½ ft per
minute. If Ela started at the surface, how far below the surface was she after 6 minutes?
(Hint: below sea level is a negative value)

Answers

Answer:

30 1/2

Step-by-step explanation:

7. (01.03 LC) A craft store marks up all sewing machines 45%. What is the selling price of a sewing machine that costs $155.00 before the markup? (1 point) $169.75 $224.75 $245.50 O $278.25

Answers

The selling price is $224.75.

What is Percent?

To determine the quantity or percentage of something in terms of 100, use the percentage formula. Per cent simply means one in a hundred. Using the percentage formula, a number between 0 and 1 can be expressed. A number that is expressed as a fraction of 100 is what it is. It is mostly used to compare and determine ratios and is represented by the symbol %.

Given:

Cost price= $155

Cost marks up = 45%

Then the selling price is

=155 + 155 x 45/100

= 155 + 155 x 0.45

=155 + 69.75

= $224.75

Hence, The selling price is $224.75.

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Write the domain and range of the function using interval notation.

Answers

The domain of the function using interval notation is 4 ≤ y < 8 and the range of the function using interval notation is 4 ≤ y < 6.

What is a domain?

A domain simply refers to the set of all real numbers for which a particular function is defined. This ultimately implies that, a domain is the set of all possible input numerical values (numbers) to a function and the domain of a graph comprises all the input numerical values which are shown on the x-axis.

By critically observing this graph of a function shown, we can reasonably infer and logically deduce the following:

Domain = (4, 8).Range = (4, 6).

In interval notation, the domain of this function can be rewritten as 4 ≤ y < 8. In interval notation, the range of this function can be rewritten as 4 ≤ y < 6.

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Use the picture shown to solve for inequalities in two triangles

Answers

GIVEN

Two triangles ABC and ACD where two corresponding sides of the two triangles are equal.

SOLUTION

The two triangles have equal sides such that:

[tex]\begin{gathered} AB=AD \\ and \\ AC=AC \end{gathered}[/tex]

By this, the length of the remaining sides BC and DC are related such that the side opposite the greater angle has the greater length.

Since [tex]BC>DC[/tex]Substitute known values into the inequality:

[tex]\begin{gathered} DC=11x-33 \\ BC=77 \\ \therefore \\ 77>11x-33 \end{gathered}[/tex]

Solving the inequality:

[tex]\:x<10[/tex]

Recall that the side DC has to be greater than 0. Therefore:

[tex]\begin{gathered} 11x-33>0 \\ \therefore \\ x>3 \end{gathered}[/tex]

Combining both solutions:

[tex]3ANSWER

What is the y-intercept of the given graph?

A) -4
B) 3
C) 4
D) None of these choices are correct.

Answers

Answer:

Answer should be 3

Step-by-step explanation:

prove me wrong

A truck rental company rents a moving truck for one day by charging $23 plus $0.09 per mile. Write a linear equation that relates the cost C, in dollars, of renting the truck to the number x of miles driven.
What is the cost of renting the truck if the truck is driven 107 miles? 419 miles?

Answers

A linear equation is an algebraic equation of the form y = mx + b. The cost of renting the truck if the truck is driven 107 miles exists $32.63.

What is meant by linear equation?

A linear equation is an algebraic equation of the form y = mx + b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."

An equation is said to be linear if the maximum power of the variable is consistently 1. Another name for it is a one-degree equation. A linear equation with one variable has the conventional form Ax + B = 0. In this case, the variables x and A are variables, while B is a constant.

Number of miles travel exists represented by "x"

cost per mile is represented by "c"

therefore we have equation,

c = 23 + 0.09 x

For 1 mile that is x = 1, cost c = $25.07

For 220 miles that is x = 220

c= 23 + 0.09 × 107

c= $32.63

For 107 miles the cost will be $32.63.

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