Find the solution to the equations.

3x - y = -4
x + y = 0

(0, 0)
(-1, 1)
(1, -1)

Answer:

It’s actually 5

Step-by-step explanation:

Ape x

The correct option is A: [tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]

To determine which option is equivalent to the given function f(x) = [tex]3(2)^{3x} \),[/tex] we will simplify the given function and compare it to each option.

The given function is:

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

We can rewrite the exponent part using properties of exponents:

[tex]\[ (2)^{3x} = (2^3)^x = 8^x \][/tex]

So the function becomes:

[tex]\[ f(x) = 3 \cdot 8^x \][/tex]

Now, let's compare this to each of the given options:

Option A: [tex]\( f(x) = 3(8)^x \)[/tex]

This is exactly the same as our simplified function. Therefore, Option A is equivalent to [tex]\( f(x) = 3(2)^{3x} \).[/tex]

Option B: [tex]\( f(x) = 24^x \)[/tex]

This is not equivalent because it implies a different base raised to the power of x.

Option C:[tex]\( f(x) = 3(8x) \)[/tex]

This is not equivalent because it multiplies by 27 instead of 3.

Option D: [tex]\( f(x) = 3(8x) \)[/tex]

This is not equivalent because it implies multiplying 8 by x, not raising 8 to the power of x.

Therefore, the correct answer is:

[tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]

The complete question is here:

Which of the following options is an equivalent function to f(x) = 3(2)3x?

A. f(x) = 3(8)x

B. f(x) = 24x

C. f(x) = 27(8)x

D. f(x) = 3(8x)

Answer:

B. Strech

Step-by-step explanation:

Stretching can change the shape of something such as a function.

It take 6.434 month  or  6 month and 13 days you to accumulate a balance of $1000 in your account.

Word problem is a mathematical problem totally described in language that is frequently used as an educational tool.

Your current account balance is $215.

Your per month income is $444.

Your per month expense is $322.

Hence, your per month savings is = $444 - $322 = $122.

Let, it take x month you to accumulate a balance of $1000 in your account.

Then mathematically,

215 + 122 x = 1000

Rearrange the equation can be written that:

122 x = 1000 -215

After the subtraction, the result will be:

122 x = 785

Rearrange the equation can be written that:

x = 785/122

After the division, the result will be:

x = 6.434 = 6 month and 13 days.

Hence,  it take 6.434 month  or  6 month and 13 days you to accumulate a balance of $1000 in your account.

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A hair dryer converts electrical energy mainly into thermal energy to heat the air, but it also produces mechanical energy to power a fan and creates sound energy during operation.

A hair dryer converts electrical energy into three main forms of energy: thermal energy, mechanical energy, and sound energy. When a hair dryer is turned on, the electrical energy is primarily converted into thermal energy to heat the air. This is achieved by passing the electric current through resistive heating coils, which transfer energy to the surrounding air as heat. Furthermore, some of the electrical energy is transformed into mechanical energy to power a fan that blows the hot air out of the dryer. Lastly, the operation of the hair dryer also produces sound energy due to the movement of the fan and the rushing air.

Answer:

Length of the rectangle = 13 ft

Step-by-step explanation:

Let l be the length and w be the width of this rectangle.

Two less than three times the width of a rectangle is equal to the length, that is

          3w - 2 = l---------------------------eqn 1

The area of the rectangle is 65 square ft

          l x w = 65  

Substituting eqn 1

           (3w - 2) x w = 65

            3w² - 2w -65 = 0

Solving quadratic eqn

            [tex]w=\frac{-(-2)\pm \sqrt{(-2)^2-4\times 3\times (-65)}}{2\times 3}=\frac{2\pm \sqrt{4+780}}{6}\\\\w=\frac{2\pm \sqrt{784}}{6}\\\\w=\frac{2\pm 28}{6}\\\\w=\frac{2+28}{6}\texttt{ or }w=\frac{2-28}{6}\\\\w=5\texttt{ or }w=-4[/tex]

w cannot be negative

Hence width, w = 5 ft

Substituting in eqn 1

              3 x 5 - 2 = l

             l = 13 ft

Length = 13 ft

Length of the rectangle = 13 ft

Answer: The correct option is A.

Explanation:

The given function is,

[tex]f(x)=2(0.4)^x[/tex]

To find the graph of this function after the reflecting across y-axis, first we have to find the graph of the equation.

The value of the function is 2 when x=0, so, the graph of given equation intersect the y-axis at 2.

In the equation [tex](0.4)^x[/tex]. Since [tex]0<0.4<1[/tex], so the given function is decreasing function.

[tex]f(x)\rightarrow 0 \text{ as }\rightarrow \infty[/tex]

[tex]f(x)\rightarrow \infty \text{ as }\rightarrow -\infty[/tex]

The value of f(x) is always positive, so the graph of f(x) is always above the x-axis. Thus, the graph must be above the x-axis after reflection across y-axis.

So, the option (2) and (4) and incorrect.

When we reflect the graph across the y-axis then,

[tex]f(x)\rightarrow \infty \text{ as }\rightarrow \infty[/tex]

[tex]f(x)\rightarrow 0 \text{ as }\rightarrow -\infty[/tex]

It means when x approaches to large negative number the f(x) approaches to 0 and when x approaches to large positive number the f(x) approaches to infinite.

Therefore, the correct option is show in first graph.

Answer:

a

Step-by-step explanation:

answer: (f-i)/-g=t

I hope it helps~

This is a problem of calculus where the product and quotient rules are used to find the derivatives of functions u and v at x=1. The values need to be taken from the provided graph.

This is a calculus question related to the product rule and the quotient rule of derivatives. First, let's explore the definitions: the product rule states that the derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second. The quotient rule states that the derivative of the quotient of two functions is the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared. From the graph, we have to find out the values of f(1), g(1), f'(1), and g'(1).

Then, we can calculate u'(1) by using the product rule: u'(1) = f'(1)g(1) + f(1)g'(1). In the same way, we can calculate v'(1) by using the quotient rule: v'(1) = [g(1)f'(1) - f(1)g'(1)] / [g(1)]^2. The specific values need to be derived from the provided graph

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(u’(1) = 5)

[tex](v’(1) = \frac{-13}{4})[/tex]

We have two functions:

(u(x) = f(x)g(x))

[tex](v(x) = \frac{f(x)}{g(x)})[/tex]

We need to find the derivatives of both (u(x)) and (v(x)) at (x = 1).

1. Derivative of (u(x)):

Using the product rule, we have:

[ u’(x) = [tex]v \cdot \frac{du}{dx} + u \cdot \frac{dv}{dx} ][/tex]

Where:

(v = g(x))

[tex](\frac{du}{dx})[/tex] is the derivative of (f(x))

[tex](\frac{dv}{dx})[/tex] is the derivative of (g(x))

From the graph:

At (x = 1), we have (f(1) = 3) and an estimated slope of the tangent line for (f’(1)[tex]\approx -2).[/tex]

Similarly, at (x = 1), we have (g(1) = 2) and an estimated slope of the tangent line for [tex](g’(1) \approx 3).[/tex]

Now let’s calculate (u’(1)):

[ u’(1) = [tex]g(1) \cdot f’(1) + f(1) \cdot g’(1)[/tex] = (2)(-2) + (3)(3) = 5 ]

2. Derivative of (v(x)):

Using the quotient rule, we have:

[ v’(x) = [tex]\frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ][/tex]

Substitute the values we found earlier:

(v = g(x) = 2)

[tex](\frac{du}{dx}[/tex] = f’(x) = -2)

[tex](\frac{dv}{dx} = g’(x) = 3)[/tex]

Now let’s calculate (v’(1)):

[tex][ v’(1) = \frac{(2)(-2) - (3)(3)}{2^2} = \frac{-13}{4} ][/tex]

Therefore:

(u’(1) = 5)

[tex](v’(1) = \frac{-13}{4})[/tex]

A factory producing 1200 pounds of chips in four hours will yield 3000 pounds when operating for 10 hours, assuming a consistent production rate.

The given problem pertains to rate, which allows the determination of how much of something can be done in a specific amount of time. In this case, the rate of potato chip production is 1200 pounds every four hours. To find out how many pounds of potato chips can be produced in 10 hours, we first need to determine the hourly rate of production.

Firstly, we determine the hourly production rate of the factory by dividing the total produce by the time spent. That is 1200 pounds / 4 hours = 300 pounds/hour.

At this hourly rate, the amount of potato chips that can be produced in 10 hours can be calculated by multiplying the hourly rate by the number of hours, which is 300 pounds/hour * 10 hours = 3000 pounds.

So, a factory that produces 1200 pounds of potato chips in four hours will make 3000 pounds of potato chips in 10 hours.

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Answer:

true

Step-by-step explanation:

The statement 'The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.' is True.

"It is a set of points that satisfy given function."

"It means that the curve representing the graph either 'moves to left / right/ up/ down' or 'it expands or compresses' or 'it reflects'."

"A shift of the graph of a  function up, down, left, or right, without changing the shape, size, or dimensions of the graph."

For given question,

We have been given a function y = 6cos x - 3

Also we have been given a transformed function y = 6cos(x - 2) - 3

We know that when function f(x) is transformed to f(x - k) then the graph of f(x) is translated 'k' units right.

So, the graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.

Therefore, the statement 'The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.' is True.

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