Function f is a rational function with a y-intercept at (0,2). if g(x) = 4f(x), which point represents the y-intercept of function g?

Answers:

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Explanation:

Problems 1, 2, and 5 are exponential functions of the form [tex]y = a(b)^x[/tex] where b is the base of the exponent and 'a' is the starting term (when x=0).

If 0 < b < 1, then the exponential function decreases or decays. Perhaps a classic example would be to study how a certain element decays into something else. The exponential curve goes downhill when moving to the right.

If b > 1, then we have exponential growth or increase. Population models could be one example; though keep in mind that there is a carrying capacity at some point. The exponential curve goes uphill when moving to the right.

In problems 1 and 5, we have b = 2 and b = 1.1 respectively. We can see b > 1 leads to exponential growth. I recommend making either a graph or table of values to see what's going on.

Meanwhile, problem 2 has b = 0.8 to represent exponential decay of 20%. It loses 20% of its value each time x increases by 1.

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Problems 3 and 4 are linear functions of the form y = mx+b

m = slope

b = y intercept

This b value is not to be confused with the previously mentioned b value used with exponential functions. They're two different things. Unfortunately letters tend to get reused.

If m is positive, then the linear function is said to be increasing. The line goes uphill when moving to the right.

On the other hand if m is negative, then we go downhill while moving to the right. This line is decreasing.

Problem 3 has a negative slope, so it is decreasing. Problem 4 has a positive slope which is increasing.

Answer: [tex]g(x)=5^x - 10[/tex]

Step-by-step explanation:

g(x) is the graph of f(x) translated 10 units down.

Answer:

see below

Step-by-step explanation:

90 degree clockwise will change x, y coordinates to  y, -x

point D  1,3    becomes   3,-1

point E   4, 4   becomes  4,-4

point F  4, -2  becomes   -2,-4

point G  1,-1    becomes    -1, -1

Step-by-step explanation:

x, y = 2 numbers

x = 2y

y - x/2 = 20

this is impossible to solve. there is no solution.

because the first equation tells us

y = x/2

and so the second equation is

x/2 - x/2 = 20

0 = 20

so, again, no solution.

Answer:

p = 21.4 cm

∠P = 44°

Step-by-step explanation:

Apply the Pythagorean Theorem to solve for the length of p.

Now, take the inverse sine function to find angle P :

Answer:

  a) 21.4 cm

  b) 45.6°

Step-by-step explanation:

The mnemonic SOH CAH TOA is intended to remind you of the relationships between trig functions and sides of a right triangle. Here, two relevant function are ...

  Cos = Adjacent/Hypotenuse

  Tan = Opposite/Adjacent

The hypotenuse of the triangle, and the side adjacent to angle P are given, so we have ...

  cos(P) = (21 cm)/(30 cm)

The angle value can be found using the inverse cosine function:

  P = arccos(21/30)

  P ≈ 45.6°

Now, we know angle P and the side adjacent to it. We want to find the measure of side p, which is opposite the angle. This can be done using either the sine function or the tangent function. Here, we choose to use the tangent function.

  tan(P) = p/21

  p = 21×tan(P) = 21·tan(45.573°)

  p ≈ 21.4 . . . . cm

__

Additional comment

In the attached, part of the calculator keypad is shown so you can see that the inverse cosine function is the 2ND function of the Cos key. The calculator mode is set to DEG (lower left of display) so that angles are in units appropriate to this problem.

Of course, you can always use the Pythagorean theorem to find the length of side p. More calculations are involved, so we used trig functions instead. Note that the angle P must be found first using the approach here, and its value must be retained with enough significant digits to ensure accuracy in the 'p' calculation.

The length of side OA in the given rhombus is: 5 units.

A rhombus is a quadrilateral that has four sides that are all of the same length. Therefore, in the image given, the length of AB equals the length of OA.

Given the following:

Length of AB = |8 - 3}

Length of AB = 5 units.

AB = OA = 5 units.

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Answer:

(7,3)

Step-by-step explanation:

you can add the equations together to eliminate y and first solve for x

(x-3y = -2) + (x+3y=16) = (2x = 14)

x = 7

then plug x into one of the equations to solve for y

(7) - 3y = -2

9 = 3y

y = 3

the solution is (7,3)

Answer: (7, 3)

Step-by-step explanation:

x - 3y = -2

x + 3y = 16

Here we can add the two equations together to cancel out the y

2x = 14

divide by 2 on both sides to get

x = 7

Now that we have the X value, plug in x value into an equation to get the Y

x - 3y = -2

7 - 3y = -2

-3y = -9

y = 3

So the solution is (7, 3)

Answer:

  -5/4

Step-by-step explanation:

To evaluate the function, put the value of the argument where the variable is and do the arithmetic.

  f(x) = -x -3x²

  f(1/2) = -(1/2) -3(1/2)² = -1/2 -3(1/4) . . . . . use 1/2 in place of x

  f(1/2) = -2/4 -3/4 = -5/4

The value in simplest form is ...

  f(1/2) = -5/4

Considering the given linear function, the inequality graphed is:

B. [tex]y \geq 2x + 2[/tex].

A linear function is modeled by:

y = mx + b

In which:

The line intersects the y-axis at 2 units, hence the y-intercept is b = 2. The function also passes through (1,4), hence the slope is:

m = (4 - 2)/(2 - 1) = 2.

Thus the equation of the line is:

y = 2x + 2.

The left-side of the line is the values above the line, hence the inequality is:

B. [tex]y \geq 2x + 2[/tex].

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Answer:

x = 3, x = 9/4

Step-by-step explanation:

8x = 24

x = 3 Divide both sides by 8.

4x = 9

x = 9/4 Divide both sides by 4.

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Equation \#1.}[/tex]

[tex]\mathsf{8x = 24}[/tex]

[tex]\huge\textbf{Simplifying:}[/tex]

[tex]\mathsf{8x = 24}[/tex]

[tex]\huge\textbf{Divide \boxed{8} to both sides:}[/tex]

[tex]\mathsf{\dfrac{8x}{8} = \dfrac{24}{8}}[/tex]

[tex]\huge\textbf{Simplify it:}[/tex]

[tex]\mathsf{x = \dfrac{24}{8}}[/tex]

[tex]\mathsf{x = 3}[/tex]

[tex]\huge\textbf{Therefore, the answer should be:}[/tex]

[tex]\huge\boxed{\mathsf{x =}\frak{\ 3}}\huge\checkmark[/tex]

[tex]\huge\textbf{Equation \#2.}[/tex]

[tex]\mathsf{4x = 9}[/tex]

[tex]\huge\textbf{Simplifying:}[/tex]

[tex]\mathsf{4x = 9}[/tex]

[tex]\huge\textbf{Divide \boxed{9} to both sides:}[/tex]

[tex]\mathsf{\dfrac{4x}{4} = \dfrac{9}{4}}[/tex]

[tex]\huge\textbf{Simplify it:}[/tex]

[tex]\mathsf{x = \dfrac{9}{4}}[/tex]

[tex]\mathsf{x = 2 \dfrac{1}{4}}[/tex]

[tex]\huge\textbf{Therefore, the answer should be:}[/tex]

[tex]\huge\boxed{\mathsf{x = }\frak{\ \dfrac{9}{4}}}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

~[tex]\frak{Amphitrite1040:)}[/tex]

we conclude that the sequence has 8 terms.

Here we have the geometric sequence:

2/81, 4/27, 8/9, ..., 6912

Notice that each term is equal to 6 times the previous term, such that:

2/81*6 = 4/27

4/27*6 = 8/9

Then the n-th term is equal to:

[tex]a_n = a_1*r^{n-1}[/tex]

Where in this case:

[tex]a_1 = 2/81\\\\r = 6[/tex]

Now we need to find the value of n such that:

[tex](2/81)*6^{n-1} = 6912\\\\6^n = 6*6812*(81/2) = 1,679,616[/tex]

If we apply the natural logarithm to both sides, then:

[tex]n*ln(6) = ln(1,679,616)\\n = ln(1,679,616)/ln(6) = 8[/tex]

Then we conclude that the sequence has 8 terms.

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Answer:

88.5

Step-by-step explanation:

3.5 * 8 = 28, this is how much was spent on copies printed. If 800 copies cost $64 and 28 of that was on the 3.5 fee per 100, then the flat rate must be:

64-28 = 36.

Now, let's do 1500. For 1500 copies, you will get charged $3.5 15 times. This amounts to 52.5, then adding the fixed fee of 36 dollars, you get

88.5

Taking 1 in the box it creates an expression equivalent to 2 - 5.

The expressions represent the distance between the two points on the

number line are |-3 -2|, |2-(-3)|, |-3 + (-2)|, |-3 - (-2)| etc.

What is integers on number line?

Integers are represented using a number line. A straight line of numbers is shown visually as a number line. It is made up of positive integers like 1, 2, etc., negative integers like - 1, - 2, etc., and a zero that is neither positive nor negative.

Part A:

2 - 5 = -3

2 + 1 = 3

The distance on the number line between a number and 0 is its absolute value.

             a. That distance is never negative.

             b. The symbol for Absolute Value is “ | | ”.

Hence, by taking 1 in the box it creates an expression equivalent to 2 - 5.

Part B:

|-3 -2| = 1

This point represent distance from -3 to -2 on the number line is 1 unit.

|2-(-3)| = 5

This point represent distance from 2 to -3 on the number line is 5 units.

|-3 + (-2)| = 5

This point represent distance from -3 to -2 on the number line is 5 units.

|-3 - (-2)| = 1

This point represent distance from -3 to -2 on the number line is 1 units.

Therefore,

Taking 1 in the box it creates an expression equivalent to 2 - 5.

The expressions represent the distance between the two points on the

number line are |-3 -2|, |2-(-3)|, |-3 + (-2)|, |-3 - (-2)| etc.

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Answer:

2x - 5 < -8 or 2x - 5 > 8

Step-by-step explanation:

|2x - 5| > 8

2x - 5 < -8 or 2x - 5 > 8

The two inequalities that can be written from the |2x - 5| > 8 are:

2x - 5 > 8

-2x + 5 > 8

We have,

To split the inequality |2x - 5| > 8 into two separate inequalities, we consider both the positive and negative cases for the absolute value expression.

Positive case:

When 2x - 5 is positive, we have:

2x - 5 > 8 _____(1)

Negative case:

When 2x - 5 is negative, we have:

-(2x - 5) > 8

Simplifying the negative case:

-2x + 5 > 8 ______(2)

Now, we have two inequalities:

2x - 5 > 8

-2x + 5 > 8

These are the two inequalities resulting from the given inequality

|2x - 5| > 8.

Thus,

The two inequalities that can be written from the |2x - 5| > 8 are:

2x - 5 > 8

-2x + 5 > 8

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At the initial condition, this weight approached the equilibrium position from a positive direction of 5 units. Also, the weight was descending (down) to the ground, with the spring stretching.

First of all, we would derive an expression for the position with respect to the angle (phase shift) under the appropriate conditions:

Asin(ωt + ø) = c₂sinωt + c₁cosωt.

At t = 0, we have:

y = c₂sin(0) + c₁cos(0).

y = 0 + c₁

y = 0 + 5

y = 5 units.

Therefore, we can logically deduce that at the initial condition, this weight approached the equilibrium position from a positive direction of 5 units. Also, the weight was descending (down) to the ground, with the spring stretching.

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Complete Question:

The angle Φ represents the phase shift, determined by the initial conditions of the experiment or the position of the weight at t = 0. If the weight is at its maximum positive position (weight is above equilibrium) at t = 0, then Φ = 0. If the weight is at its maximum negative position (spring is stretched and weight is below equilibrium) at t = 0, then Φ = π. If the weight is traveling in the negative direction and passing through equilibrium at t = 0, then Φ = π/2. In the response box, describe the initial condition of our experiment; specifically, describe the position of the weight and the direction in which it was traveling.

The simplified equation of [tex](x - 10)^2 + (y- 6)^2 = 4^2[/tex] is [tex]y = 6+ \sqrt{16 - (x - 10)^2}[/tex]

The complete question is added as an attachment

The equation is given as

[tex](x - 10)^2 + (y- 6)^2 = 4^2[/tex]

Subtract (x - 10)^2 from both sides

[tex](y- 6)^2 = 4^2 - (x - 10)^2[/tex]

Evaluate 4^2

[tex](y- 6)^2 =16 - (x - 10)^2[/tex]

Take the square root of  both sides

[tex]y- 6= \sqrt{16 - (x - 10)^2}[/tex]

Add 6 to both sides

[tex]y = 6+ \sqrt{16 - (x - 10)^2}[/tex]

Hence, the simplified equation of [tex](x - 10)^2 + (y- 6)^2 = 4^2[/tex] is [tex]y = 6+ \sqrt{16 - (x - 10)^2}[/tex]

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Answer:

1. pi.      2.49-12.25pi

Step-by-step explanation:

The value of the 4th iterate of the given function f(x) is -2.096

2nd option is correct.

A function's iterate is a function that is expressed as repetition of another (already iterated) function; this latter function may be referred to as the iterand.

Any function taken on its own is regarded as the first iteration.

A function is said to have an identity function at iteration zero.

According to the question,

f(x) = (3x-1)/[tex]x^{2}[/tex]

[tex]x_{o}[/tex] = -2

The first iterate is calculated by finding f(-2).

f(-2) = (3(-2)-1)/4

f(-2) = -1.75

Similarly, the next iterates are calculated as follows:

f(-1.75) = -2.048

f(-2.048) = -1.71

f(-1.71) = -2.096

Thus, the fourth iterate for the given function is [tex]x_{4}[/tex]= -2.096

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Answer:

[tex] [/tex] [tex] [/tex] [tex] [/tex] [tex] [/tex]

The height of a pyramid with volume of 75 cm³ and square base area of 25 cm², is 9 cm

An equation is an expression that shows the relationship between two or more variables and numbers.

Let us assume that the pyramid has a volume of 75 cm³ and square base area of 25 cm², hence:

Volume = (1/3) * area of base * height

75 = (1/3) * 25 * height

height = 9cm

The height of a pyramid with volume of 75 cm³ and square base area of 25 cm², is 9 cm

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Answer:

b) x = 4

Explanation:

[tex]2x^2 - x + 10 = 38[/tex]

collect variables

[tex]2x^2 - x + 10 - 38 = 0[/tex]

simplify

[tex]2x^2 - x - 28 = 0[/tex]

breakdown

[tex]2x^2 -8x + 7x - 28 = 0[/tex]

factor out

[tex]2x(x - 4) + 7(x - 4) = 0[/tex]

collect into groups

[tex](2x + 7)(x - 4) = 0[/tex]

set to zero

[tex]2x + 7 = 0, x - 4 = 0[/tex]

relocate variable

[tex]2x = -7, x = 4[/tex]

final solutions

[tex]x = -3.5, x = 4[/tex]

The answer is b.

To find the value of x, we need to convert this into the form of a quadratic equation.

Subtract 38 from each side.

Solve by splitting the middle term.

The surface area of the composite solid is 579.84 square feet

The composite solid that completes the question is added as an attachment

From the attached figure, we have the following figures:

The surface area of a rectangular prism is

A = 2(lw + lh + wh)

Using the dimensions of the prism, we have:

A = 2(6 * 11 + 6 * 8 + 8 * 11)

This gives

A = 404

The surface area of a cylinder is

A = 2πr^2 + 2πrh

Using the dimensions of the cylinder, we have:

A = 2 * 3.14 * 4^2 + 2 * 3.14 * 4 * 5

Evaluate

A = 226.08

The area of the circle between both shapes is

A = πr^2

This gives

A = 3.14 * 4^2

A = 50.24

The surface area of the composite solid is then calculated as:

Surface area = Area of rectangular prism + Area of cylinder - Area of circle

This gives

Surface area = 404 + 226.08 - 50.24

Evaluate

Surface area = 579.84

Hence, the surface area of the composite solid is 579.84 square feet

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Probability that a card selected at random is purple = 7/25

Number of red cards, N(red) = 12

Number of blue cards, N(blue) = 17

Number of purple cards, N(purple) = 14

Number of yellow cards, N(yellow) = 7

Total number of cards, N(total) = 12 + 17 + 14 + 7

N(total) = 50

Probability that a card selected at random will be purple

P(purple) = N(purple) / N(total)

P(purple) = 14/50

Reduce to the lowest fraction

P(purple) = 7/25

Therefore, probability that a card selected at random is purple = 7/25

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Complete Question

There are 12 red cards, 17 blue cards, 14 purple cards, and 7 yellow cards in a hat. If a card is selected at random, what is the probability that the card is purple

The area and circumference of a circle are 12.56square feet and 12.56ft respectively

The formula for calculating the area and circumference of a circle is expressed as:

A = πr²

C = πd

where

r is the radius and d is the diameter

Area = 3.14 * (2)²

Area = 12.56ft²

For the circumference

C = 3.14(4)

C = 12.54ft

Hence the area and circumference of a circle are 12.56square feet and 12.56ft respectively

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Answer:

d. 64

Step-by-step explanation:

x^1/2 = 8

Square each side

(x^1/2 )^2=( 8)^2

x = 64

Answer:

x=1

Step-by-step explanation:

1 + 1/3 = 4/9

1/3 also equals 3/9

hope this helped :)

Answer :

Step-by-step explanation :

>> (3 × x + x / 3) = 4 / 9

>> (3x + x / 3) = 4 / 9

>> (4x / 3) = 4 / 9

>> 4x × 9 = 3 × 4

>> 36x = 12

>> x = 12 / 36

>> x = 6 / 18

>> x = 3 / 9

>> x = 1 / 3

The ramp with the following dimension won't obey the ADA guideline because the angle is more than 4.8 degrees.

The angle of elevation of the ramp can be found using trigonometric ratios.

Therefore,  the angle of elevation must be equal to 4.8 degrees to obey the ADA guidelines .

Hence,

tan ∅ = opposite / adjacent

opposite side = 3.5 ft

adjacent side = 40 ft

Therefore,

tan ∅ = 3.5 / 40

∅ = tan ⁻¹ 0.0875

∅ = 5.00064459756

∅ = 5 degrees

Therefore, the ramp with the following dimension won't obey the ADA guideline because the angle is more than 4.8 degrees.

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