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3 years ago

What is the domain of the step function f(x) = ⌈2x⌉ – 1?

Mathematics

2 answers:

3241004551 [841]3 years ago

5 0

F(x) is defined for "all real numbers."

Damm [24]3 years ago

5 0

<h2>Answer:</h2>

The domain of the given step function is:

The set of all the real numbers.

<h2>Step-by-step explanation:</h2>

<u>Ceiling function--</u>

The ceiling function also known as the least integer function is the function which takes the largest value of the integer for a non-integer number which lie between the two integers , and gives the integer itself if the number is integer.

and it is given by:

$f(x)=\left \lceil x \right \rceil$

(

For example--

if x= 0.5

then f(x)=1 since x lie between 0 and 1

if x= -2.5

Then f(x)= -2

since x lie between -3 and -2.

if x=3

then f(x)=3 )

Also, the function is defined for all real values of x.

i.e. The domain of such a function is: All real numbers.

The function f(x) is given by:

$f(x)=\left \lceil 2x \right \rceil-1$

Hence, the domain is all the real numbers.

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Share 66 g of sugar in the ratio 2:3:1?

slega [8]

Answer:

whaaaaaaaaaattt????

Step-by-step explanation:

6 0

2 years ago

Solve for x 0=3x^2+3x+7

Aloiza [94]

Answer:

x =(3-√-75)/-6=1/-2+5i/6√ 3 = -0.5000-1.4434i

x =(3+√-75)/-6=1/-2-5i/6√ 3 = -0.5000+1.4434i

Step-by-step explanation:

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

0-(3*x^2+3*x+7)=0

Step by step solution:

Step 1:

Equation at the end of step 1 :

0 - (( $-3x^{2}$ + 3x) + 7) = 0

Pulling out like terms:

2.1 Pull out like factors:

$-3x^{2}$ - 3x - 7 = -1 • ( $3x^{2}$ + 3x + 7)

Trying to factor by splitting the middle term

2.2 Factoring $3x^{2}$ + 3x + 7

The first term is, $3x^{2}$ its coefficient is 3 .

The middle term is, +3x its coefficient is 3 .

The last term, "the constant", is +7

Step-1 : Multiply the coefficient of the first term by the constant 3 • 7 = 21

Step-2 : Find two factors of 21 whose sum equals the coefficient of the middle term, which is 3 .

-21 + -1 = -22

-7 + -3 = -10

-3 + -7 = -10

-1 + -21 = -22

1 + 21 = 22

3 + 7 = 10

7 + 3 = 10

21 + 1 = 22

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Equation at the end of step 3 :

$-3x^{2}$ - 3x - 7 = 0

Parabola, Finding the Vertex:

3.1 Find the Vertex of y = $-3x^{2}$ -3x-7

For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is -0.5000

Plugging into the parabola formula -0.5000 for x we can calculate the y -coordinate :

y = -3.0 * -0.50 * -0.50 - 3.0 * -0.50 - 7.0

or y = -6.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = $-3x^{2}$ -3x-7

Axis of Symmetry (dashed) {x}={-0.50}

Vertex at {x,y} = {-0.50,-6.25}

Function has no real roots

Solve Quadratic Equation by Completing The Square

3.2 Solving $-3x^{2}$ -3x-7 = 0 by Completing The Square .

Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:

$3x^{2}$ +3x+7 = 0 Divide both sides of the equation by 3 to have 1 as the coefficient of the first term :

$x^{2}$ +x+(7/3) = 0

Subtract 7/3 from both side of the equation :

$x^{2}$ +x = -7/3

Now the clever bit: Take the coefficient of x , which is 1 , divide by two, giving 1/2 , and finally square it giving 1/4

Add 1/4 to both sides of the equation :

On the right hand side we have :

-7/3 + 1/4 The common denominator of the two fractions is 12 Adding (-28/12)+(3/12) gives -25/12

So adding to both sides we finally get :

$x^{2}$ +x+(1/4) = -25/12

Adding 1/4 has completed the left hand side into a perfect square :

$x^{2}$ +x+(1/4) =

(x+(1/2)) • (x+(1/2)) =

(x+(1/2))2

Things which are equal to the same thing are also equal to one another. Since

$x^{2}$ +x+(1/4) = -25/12 and

$x^{2}$ +x+(1/4) = (x+(1/2))2

then, according to the law of transitivity,

(x+(1/2))2 = -25/12

We'll refer to this Equation as Eq. #3.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(x+(1/2))2 is

(x+(1/2))2/2 =

(x+(1/2))1 =

x+(1/2)

Now, applying the Square Root Principle to Eq. #4.2.1 we get:

x+(1/2) = √ -25/12

Subtract 1/2 from both sides to obtain:

x = -1/2 + √ -25/12

√ 3 , rounded to 4 decimal digits, is 1.7321

So now we are looking at:

x = ( 3 ± 5 • 1.732 i ) / -6

Two imaginary solutions :

x =(3+√-75)/-6=1/-2-5i/6√ 3 = -0.5000+1.4434i

or:

x =(3-√-75)/-6=1/-2+5i/6√ 3 = -0.5000-1.4434i

6 0

3 years ago

Find the volume of the rectangular prism. Write your answer as a fraction.

suter [353]

Answer:

V = 8/125 cubic units

Step-by-step explanation:

This prism is actually a cube of side length 2/5.

The volume of a cube of side length s is V = s^3.

Here, the volume of the cube (rectangular prism) is V = (2/5)^3, or

V = 8/125 cubic units

8 0

2 years ago

it is known that the population proton of utha residnet that are members of the church of jesus christ 0l6 suppose a random samp

Lady_Fox [76]

Answer:

0.0838 = 8.38% probability of obtaining a sample proportion less than 0.5.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$