I need help on the back side

The function [tex]\( f(x) = \frac{5x}{x-x^2} \)[/tex] has a removable discontinuity at ( x = 0 ) after canceling the common factor [tex]\( x \).[/tex]

A removable discontinuity occurs at a point where a function is not defined due to a factor in the denominator that could be canceled out with a factor in the numerator.

Let's analyze each function to determine if any of them have a removable discontinuity.

[tex]a. \( g(x) = \frac{2x-1}{x} \)[/tex]

- The denominator ( x ) is zero at ( x = 0 ).

- The numerator ( 2x-1 ) is non-zero at ( x = 0 ).

- Since there is no common factor in the numerator and denominator that could cancel out, the discontinuity at ( x = 0 ) is not removable.

[tex]b. \( p(x) = \frac{x+2}{x^2-x-2} \)[/tex]

- The denominator [tex]\( x^2-x-2 \) factors as \( (x-2)(x+1) \).[/tex]

- The function becomes [tex]\( p(x) = \frac{x+2}{(x-2)(x+1)} \).[/tex]

- The denominator is zero at ( x = 2 ) and ( x = -1 ).

- The numerator ( x+2 ) is zero at ( x = -2 ).

- There is no common factor between the numerator and the denominator that could be canceled out. So, the discontinuities at [tex]\( x = 2 \) and \( x = -1 \)[/tex] are not removable.

[tex]c. \( f(x) = \frac{5x}{x-x^2} \)[/tex]

- The denominator [tex]\( x-x^2 \) factors as \( x(1-x) \).[/tex]

- The function becomes [tex]\( f(x) = \frac{5x}{x(1-x)} = \frac{5x}{x-x^2} \).[/tex]

- The denominator is zero at x = 0  and  x = 1 .

- The numerator ( 5x ) is zero at ( x = 0 ).

- There is a common factor of ( x ) in the numerator and denominator which could be canceled out, making the discontinuity at ( x = 0 ) removable.

- After canceling the common factor ( x ), the function becomes [tex]\( \frac{5}{1-x} \),[/tex] which is defined at ( x = 0 ).

[tex]d. \( h(x) = \frac{x^2 - x + 2}{x + 1} \)[/tex]

- The denominator ( x + 1 ) is zero at ( x = -1 ).

- The numerator [tex]\( x^2 - x + 2 \) is not zero at \( x = -1 \).[/tex]

- There is no common factor in the numerator and denominator that could be canceled out, so the discontinuity at ( x = -1 ) is not removable.

Conclusion

The function [tex]\( f(x) = \frac{5x}{x-x^2} \)[/tex] has a removable discontinuity at ( x = 0 ), because the discontinuity can be removed by canceling the common factor ( x ) in the numerator and the denominator.

So, the correct answer is:

[tex]c. \( f(x) = \frac{5x}{x-x^2} \)[/tex]

To estimate the number of grains of sand that fit in a 5-gallon bucket, we can calculate the volume of the bucket and the volume of a grain of sand. By assuming the grains are approximately the same size and shape, we can use the formula for the volume of a sphere to estimate the number of grains.

To answer this question, we need to make some assumptions. Let's assume that the grains of sand are roughly the same size and shape. We can estimate the number of grains of sand that fit in a 5-gallon bucket by considering the volume of the bucket and the volume of a grain of sand. The volume of a grain of sand can be approximated as a sphere.

Using the formula for the volume of a sphere (V = (4/3)πr³), we can find the volume of a grain of sand. If the grain of sand has sides that are 1.0 mm long, the radius of the sphere would be 0.5 mm (half of the side length).

Now, we can calculate the volume of the 5-gallon bucket, convert it to cubic millimeters, and divide it by the volume of a grain of sand to estimate the number of grains that fit in the bucket.

A 5-gallon bucket can hold approximately 36 million grains of sand.

To estimate the number of grains of sand that can fit into a 5-gallon bucket, we need to start with the volume of the bucket and an average sand grain.

A standard 5-gallon bucket is approximately 18.927 liters (since 1 gallon = 3.78541 liters).

An average grain of sand has a diameter ranging from 0.063 mm to 2 mm. We'll use an average grain size of 1 mm for our calculations. The volume V of a sphere (which we can use to approximate a sand grain) is given by the formula:

V = 4/3 π r³

For a grain of sand with a 1mm diameter, the radius r is 0.5 mm or 0.0005 meters. Therefore:

V = 4/3 π (0.0005)³ ≈ 5.24 x 10-10 m³

The volume of the 5-gallon bucket in cubic meters is:

18.927 liters = 0.018927 m³

Dividing the volume of the bucket by the volume of a single grain of sand gives us:

Number of grains = 0.018927 / 5.24 x 10-10 ≈ 3.613 x 107 grains

Therefore, approximately 36 million grains of sand can fit into a 5-gallon bucket.

What applies here is one of the laws of the factorization of polynomials, called the factor theorem and it states that:

For a polynomial f(x) , if for any value a,  f(a) =0  then (x-a) is factor of f(x)

Example:

Consider the polynomial

[tex]f(x) = x^{3} - 3x^{2} - 8x+4[/tex]

For  a =3,

[tex]f(a) = (3)^{3} - 3(3)^{2} - 8(3)+24[/tex]

= 27-27-24+24 = 0

f (3)= 0

which means (x-3) is a factor of f(x)

Applying the above rule to the question:

if (x + 3) is a factor of a polynomial f(x), then f(-3) = 0

note that (x+3) can also be written  as (x- (-3)).

Answer:

Prime

Step-by-step explanation:

Its factors are:1 and itself.

ALL prime numbers have the same 2 factors: 1 and itself.

Answer:

0.5 or half of it (1/2)

Researchers can use b) weighting to counter bias in their sampling, which adjusts sample representation to match a desired population. While convenience sampling offers easy data collection, it lacks generalizability, and weighting is necessary to enhance the validity of findings.

To counter bias in their sampling, researchers can employ a method known as b) weighting. This technique adjusts the representation of samples to match a desired population, countering potential biases that may arise from non-random sampling methods such as convenience sampling. It is important to distinguish between non-random sampling methods that are potentially biased and strategies that are used to ensure data accuracy and representativeness.

Convenience sampling, also referred to as availability sampling or haphazard sampling, is a nonprobability sampling strategy where researchers collect data from individuals or elements that are most easily accessible. This approach is useful in exploratory research or student projects where probability sampling is too costly or difficult but does not offer the rigor needed to make generalizations to larger populations. To overcome these limitations and enhance the validity of their findings, researchers may adjust their data using weighting to better reflect the population being studied.

Answer:

The answer is x=-3.

Step-by-step explanation:

I'm not sure how to word it, but I did this question on khan, and got this answer and it was right.

Answer:

m∠3=115°.

Step-by-step explanation:

It is given from the figure that line q is parallel to s and r is the transversal.

Since, m∠6 and m∠8 forms a linear pair as they are on the straight line r, therefore using the linear pair property, we have

m∠6+m∠8=180°

⇒[tex]2x-5+x+5=180^{\circ}[/tex]

⇒[tex]3x=180^{\circ}[/tex]

⇒[tex]x=60^{\circ}[/tex]

Thus, the measure of ∠6 is [tex]2x-5=2(60)-5=120-5=115^{\circ}[/tex]

Now, m∠3=m∠6=115° as both m∠3 and m∠6 forms the alternate interior angle pair.

Therefore, the measure of m∠3=115°.

Final answer:

Using the compound interest formula, the total amount after three years for $50 invested at 6% compounded monthly is approximately $59.70.

Explanation:

To calculate the future value of $50 invested at 6% compounded monthly after a period of three years, we use the formula for compound interest:

FV = P × (1 + (r/n))³⁼ ×⁴

where:

FV is the future value of the investment,

P is the principal amount ($50),

r is the annual nominal interest rate (6% or 0.06),

n is the number of times the interest is compounded per year (12, for monthly compounding),

t is the time the money is invested for, in years (3 years).

Using these values, we calculate as follows:

FV = $50 × (1 + (0.06/12))³·×´ = $50 × (1 + 0.005)·¹·´

Now we calculate this raised to the power of 36 (3 years times 12 months):

FV = $50 × (1.005)³···¶ = $50 × 1.194052

Thus, FV ≈ $59.70

The total amount after three years would be approximately $59.70, assuming the interest is compounded monthly at a rate of 6%.

D. Y=-(3+x^2)

A. Y=2x-x^2

Answer:

The answers are a) y=2x-x^2 and d) y=-(3+x^2) and in the attached file are the graphs.

Step-by-step explanation:

Quadratic equations are those where the exponent of the unknown term is squared, that is, the unknown is elevated to exponent 2. They have the general form of a trinomial:

ax2 + bx + c = 0

where a, b and c are real numbers and are called coefficients. Thus, a is the coefficient of x2, b is the term or coefficient of x and c is the independent term.

The barrel leaks 0.5 L of water each day.

Every year, Luisa puts $10 into her savings account.

Linear functions represent situations with a constant rate of change. Saving $10 every year and a barrel leaking 0.5 L of water each day are examples of linear relationships due to this constant change.

Situations that can be represented by a linear function are those where the rate of change remains constant. Let's analyze the given situations according to this condition:

These two scenarios fit the criteria for being linear because they involve a constant rate of change over time. On the other hand, scenarios like the exponential growth of bacteria or percentage-based population growth are not linear since the rate of change is not constant; these would be represented by exponential functions.

Answer:

value of [tex](f o g)(1)[/tex] is, 17

Step-by-step explanation:

We have to find the [tex](f o g)(1)[/tex]

Using the given tables;

[tex](f o g)(1) = f(g(1))[/tex]   ......[1]

At x = 1

g(1) = 6

Substitute this in [1] we have;

[tex](f o g)(1) = f(6)[/tex]

At x = 6

f(6) = 17

then;

[tex](f o g)(1) = 17[/tex]

Therefore, the value of [tex](f o g)(1)[/tex] is, 17

The points that are solutions to the system of inequalities x + y 5 + 2 y > 2 below are (5, 2),  (3, 6), (1, 1). Options B, C, A, and D. For is mathematically given as

Generally, inequalities are simply defined as  the relationship between two non-equal expressions using a symbol like "not equal to," "greater than," or "less than."

In conclusion, The points (5, 2), and (2, 5), (1, 1) are the ones that are the solutions to the system of inequalities x + y 5 + 2 y > 2 below (3, 6).

For (5, 2),

5 + 2* 5 + 2 *2 > 2

19>2

For  (2, 5)

2 + 5* 5 + 2*5 > 2

37>2

For (3, 6).

3 + 6*5 + 2*6 > 2

45>2

For (1, 1)

1 + 1* 5 + 2*1 > 2

8>2

For (1, -1)

1 + -1* 5 + 2 *-1 > 2

-6<2

For (2, -5)

2 + -5* 5 + 2*-5 > 2

−33<2

Read more about inequalities  

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Answer: The correct option are A, B and D.

Explanation:

The law of sine states that,

[tex]\frac{\sin A}{a} =\frac{\sin B}{b}=\frac{\sin C}{c} [/tex]

Where A, B, C are interior angles of the triangle and a, b, c are sides opposite these angles respectively as shown in below figure.

Since we need the combination of two angles and one side or two sides and one angle.

The Law of sine is useful for AAS and SSA type problems.

Reason for correct option:

In  option A three sides are known but no angle is not given, therefore the SSS problem can not be solved by Law of sine and the option A is correct.

In option B a side is known and two inclined angle on that line are known. But to use Law of sine we want the line and angle which in not inclined on that line, therefore the ASA problem can not be solved by Law of sine and the option B is correct.

In option D two sides and their inclined angle is known. But to use Law of sine we want the side and angle which in not inclined on that line, therefore the SAS problem can not be solved by Law of sine and the option D is correct.

Reason for incorrect option:

In option C, the two consecutive angles are given and a side which makes the second angle with base side, therefore the first angle is opposite to the given side, so the law of sine can be used for AAS problems.

Therefore, option C is incorrect.

Answer:1250

Step-by-step explanation:

Answer:  The correct option is (B) [tex]f>5.[/tex]

Step-by-step explanation:  Given that Blake has more than five friends.

We are to select the inequality that describes the number of Blake's friends.

The number of Blake's friends is represented by f.

According to the given information, we have

the inequality that describes the number of Blake's friends is given by

[tex]f>5.[/tex]

Thus, (B) is the correct option.

To find the inverse of the function y = x² - 7, interchange x and y to get x = y² - 7, then solve for y to get the inverse function y = (x + 7), taking into account both the positive and negative square roots.

To find the inverse of the function y = x² − 7, we need to swap the x and y variables and then solve for y. Here's a step-by-step process:

This results in the inverse function y = ±√(x + 7), but typically, we only take the principal square root for the inverse function, which would be y = √(x + 7) assuming x ≥ -7.