Given $f(x) = \frac{\sqrt{2x-6}}{x-3}$, what is the smallest possible integer value for $x$ such that $f(x)$ has a real number value? Please show steps. Thank you!

(I rewrote the question without the symbols, they are the same question)

Given f(x) = {2x-6}/{x-3}, what is the smallest possible integer value for x such that f(x) has a real number value? Thank you!​

Answers

Answer 1
Answer:   x = 4

===========================================================

Explanation:

The given function is

[tex]f(x) = \frac{\sqrt{2x-6}}{x-3}[/tex]

which is the same as writing f(x) = ( sqrt(2x-6) )/(x-3)

The key for now is the square root term. Specifically, the stuff underneath. This stuff is called the radicand.

Recall that the radicand cannot be negative, or else the square root stuff will result in a complex number. Eg: [tex]\sqrt{-4} = 0+2i[/tex]

The question is basically asking: what is the smallest x such that [tex]\sqrt{2x-6}[/tex] is a real number?

Well if we made 2x-6 as small as possible, ie set it equal to 0, then we can find the answer

[tex]2x-6 = 0\\\\2x = 6\\\\x = 6/2\\\\x = 3\\\\[/tex]

I set the radicand equal to 0 because that's as small as the radicand can get (otherwise, we're dipping into negative territory).

So 2x-6 set equal to 0 leads to x = 3.

This means x = 3 produces the smallest radicand (zero) and therefore, it is the smallest allowed x value for that square root term.

But wait, if we tried x = 3 in f(x), then we get...

[tex]f(x) = \frac{\sqrt{2x-6}}{x-3}\\\\f(3) = \frac{\sqrt{2*3-6}}{3-3}\\\\f(3) = \frac{\sqrt{0}}{0}\\\\[/tex]

which isn't good. We cannot have 0 in the denominator. Dividing by zero is not allowed. The result is undefined. It doesn't even lead to a complex number. So we'll need to bump x = 3 up to x = 4. You should find that x = 4 doesn't make the denominator 0.

----------------

In short, we found that x = 3 makes the square root as small as possible while staying a real number, but it causes a division by zero error with f(x) overall. So we bump up to x = 4 instead.


Related Questions

This is a list of the heights ( each nearest cm ) of 12 children
150 134 136 139 131 141
132 134 136 137 150 146
Select the type of the data.PLEASE HELP CHOOSE ONE


Discrete

Continuous

Categorical

Qualitative

Answers

Answer:

qualitative

Step-by-step explanation:

bcos it is in quality format

Evaluate 3|x|+2x-1 when x = -5.

Answers

Answer:

4

Step-by-step explanation:

To start off, we are going to input our x value into our expression.

3|-5| + 2(-5) - 1

Next, we are going to find the absolute value (always positive) of -5 and multiply 2 and -5.

3(5) - 10 -1

Now, we will multiply 3 and 5.

15 - 10 -1

Finally, we are going to combine our like terms (15, -10, and -1)

4

So! Our final answer for this expression is 4!

Hope this Helps! :)

Have any questions? Ask below in the comments and I will try my best to answer.

-SGO

Craig and Cindy working together can mow the lawn in four hours working alone Cindy takes twice as long as Craig how long does it take Craig to mow the lawn alone

Answers

Answer:

12 hours

Step-by-step explanation:

1 : (1/4 : (2 + 1)) = 12

show that the line x-y+2=0 and the line joining the points (4,6) and (10,12) are parallel to each other

Answers

x-y+2=0

-y= -x-2

y=x+2

In order for the other line to be parallel, the slope has to be the same.

(12-6)/(10-4)= 6/6= 1

Both slopes are the same, meaning it is parallel.

At a restaurant, two burgers and one fries cost 6. 50. What is the cost of six burgers and three fries

Answers

Answer:

19.5

Step-by-step explanation:

let burger be x and fries be y

2x+y = 6.5

6x+3y = 3(2x+y) =3(6.5) =19.5

answer the question ​

Answers

Answer:

g(1)=(1-1)*2 -2

= -2

g(2) = (2-1)*2 -2

= -1

g(3) = 1/2 (3) -2

= -1/2

Answer:

g(1) = -2

g(2) = -1

g(3) = -1/2 or -0.5

Step-by-step explanation:

you really need help for typing numbers into your calculator ? because there is nothing else to do here.

we get 3 values of x and need to calculate the functional values based on the given expressions by using each given value for x.

the only thing that requires a bare minimum of intelligence is to find the fitting expression.

so, for g(1) and g(2) we need to use the middle one, and for g(3) the third one. uhhhh, very difficult ...

typing it in here is way more effort than doing it directly on the calculator or really in the head.

so, g(1) = (1-1)² - 2 = -2

g(2) = (2-1)² - 2 = 1 - 2 = -1

g(3) = 1/2 ×3 - 2 = 3/2 - 2 = 3/2 - 4/2 = -1/2

uhhhh, come in, you could have done that too. and way faster.

A noted psychic was tested for extrasensory perception. The psychic was presented with 200 cards face down and asked to determine if each card were one of five symbols: a star, a cross, a circle, a square, or three wavy lines. The psychic was correct in 50 cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Assume the 200 trials can be treated as a simple random sample from the population of all guesses the psychic would make in his lifetime. How large a sample n would you need to estimate p with a margin of error of 0.01 with 95% confidence? Use the hypothesized value p = 0.20 as the value for p*.

Answers

Answer:

r3jehejn wbbwbwbbwmwkwkwjwjwhhejehehehhe

Production possibilities curve graph

Answers

Answer:

?????????????????:))

If you wanted to make a game where you pay $5 if you can't guess a random dogs weight within 16lbs what payout should you offer you make the game zero-expected value

Answers

Answer:

Following are the solution to the given question:

Step-by-step explanation:

The population std. dev of the dog weight=8

[tex]\sigma=8\\\\P(\text{guess with in 16 lbs}) = P(|X-\mu|\leq 16)\\\\=P(-2 \leq Z \leq 2) = 0.9544\\\\[/tex]

Calculating the payout w s.t:

[tex]E[netpay]=0=(-5) \times 0.9544+w\times (1-0.9544)\\\\ w =(5 \times \frac{0.9544}{1-0.9544}) =\$ 104.65[/tex]

therefore, we assume that the weight of the dog is a normal distribution with std. deviation that is 8.

What is x: |3x-1| = 4

Answers

Step-by-step explanation:

3x-1=4

3x=4+1

3x=5

x=5/3

Answer:

x = 5/3                       x = -1

Step-by-step explanation:

|3x-1| = 4

There are two solutions, one positive and one negative

3x-1 =4    and 3x-1 = -4

Add 1 to each side

3x-1+1 = 4+1     3x-1+1 = -4+1

3x = 5                  3x = -3

Divide by 3

3x/3 = 5/3             3x/3 = -3/3

x = 5/3                       x = -1

Convert 37.5% to a fraction. (Reduce your answer to lowest terms.)

Answers

Answer:

3/8

Step-by-step explanation:

The value of 37.5% reduced to the lowest terms is 15/40.

What is a fraction?

A fraction is written in the form p/q, where q ≠ 0. Fractions are of two types they are proper fractions and improper fractions. Fractions can also be converted into percentages by multiplying them by 100.

We know e can convert percentages into decimals and fractions by dividing the percentage value by hundred.

Given we have to convert 37.5% as a fraction which is,

= 37.5/100.

= 375/1000.

= 75/200.

= 15/40.

learn more about percentages here :

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Which of the following describes a positive correlation?

As the number of hours spent on homework increases, the tests scores increase.

As the number of apples eaten per year increases, the number of times visiting the doctor every year remains the same.

As the number of times going to bed early increases, the number of times waking up late decreases.

The amount of time a team spent practicing increases, the number of games lost in a season decreases.
THIS IS A MULTIPLE CHOICE QUESTION

Answers

Answer:

First Choice: As the number of hours spent on homework increases, the tests scores increase.

Step-by-step explanation:

The definition of a positive correlation  is a relationship between two given variables, in which both variables are moving in the same direction. This can mean when one variable increases and the other variable increases, too, or one variable decreases and the other decreases as well.

The first choice is a positive correlation because both variables are changing (increasing) in the same direction. As you spend more time on homework, you're likely to get a higher test score.

The second choice cannot be a positive correlation because only one variable is having some kind of change (increasing). The doctor visits amount remains the same, so we can call this a zero-correlation relationship because the number of apples eaten yearly doesn't affect the amount of doctor visits. An apple a day keeps the doctor a way is just a proverb, not to be taken literally.

The third choice cannot be a positive correlation because the two variables are going different directions. Even though the number of times going to bed early is increasing, the number of times waking up late decreases, which is not moving in the same direction as the other variable.

The fourth choice cannot be a positive correlation because, similarly to the third choice, the two variables are going different directions. One variable is increasing, which is the amount of practice time. Meanwhile, the other variable is decreasing (going in the opposite direction), which is the number of games lost in a season.

given the series 1+2+3+4+5+6+...+5000. Write the series in sigma notation if all the powers of 4 are removed from the series.​

Answers

We have 4⁶ = 4096 and 4⁷ = 16,384, which is to say that the given sum only contains the first six powers of 4.

Now,

[tex]\displaystyle 1+2+3+\cdots+5000 = \sum_{k=1}^{5000}k[/tex]

and you subtract the sum of the first six powers of 4 to get the sum S that you want,

[tex]\displaystyle S = \boxed{\sum_{k=1}^{5000}k - \sum_{k=1}^64^k}[/tex]

Referring to the scale, 1:50 what is the actual measurement of the wall marked 18 cm on the plan?

Answers

Answer:

900 cm

Step-by-step explanation:

1 : 50

We need the first number to be 18 so multiply by 18

1:18 : 50*18

18 cm : 900 cm

What is the volume of the
cylinder? Use 3.14 for a.
A 1200 cubic inches
B 1884 cubic inches
C 3768 cubic inches
D 28,260 cubic inches

Answers

Answer:

Step-by-step explanation:

area of top face = π15² = 225π in²

volume = 225π × 40 = 9000π ≅28,260 in³

please help me with this

Answers

4x-4+12=20°+35°4x+8=55°4x=55°-84x=47°x=47÷4x=11.75x=12

or

4x-4=124x=12+44x=16x=16÷4x=4

please mark this answer as brainlist

What is the equation

Answers

Answer:

D.) y = 2x + 2

Step-by-step explanation:

First, we need to find the slope.

Lets use the points (0, 2) and (-2, -2).

Using the formula for calculating slope, we get 2 as our slope.

Since the equation should be in slope-intercept form, we use this formula.

y = mx + b

We'll use our first point (0, 2) to substitute for x and y and use 2 to substitute for m (slope):

2 = 2(0) + b

2 x 0 = 0

2 = 0 + b

-0 = -0

= 2 = b

Now, substitute b for 2 for 2 for m.

= y = 2x + 2.

Hope this helps!

If there is something wrong, please let me know.

y = 95°, find the measure of x

Answers

9514 1404 393

Answer:

  x = 100°

  y = 95°

Step-by-step explanation:

It is probably easier to find y first. Opposite angles of an inscribed quadrilateral are supplementary, so ...

  y = 180° -85° = 95°

The measure of an arc is double the measure of the inscribed angle subtending it. The arc subtended by angle y is ...

  90° +x = 2y

  x = 190° -90° = 100°

_____

Additional comment

The rule cited above regarding opposite angles of an inscribed quadrilateral comes from the theorem regarding inscribed angles. In the given diagram, the diagonal from the bottom vertex to the top one is a chord that divides the circle into two arcs. Their sum is 360°. The inscribed angle theorem tells you ...

  2y +2(85°) = 360°

  y + 85° = 180° . . . . . . . divide by 2; opposite angles are supplementary

If y = sin 3x, find y" in terms of y.

Answers

Answer:

-0.54

Step-by-step explanation:

Hope it helped.

• • •

What is the area of my lawn if it measures 4m x 4.5m

Answers

Answer:

18m squared

Step-by-step explanation:

4m x 4.5m = ?

18m square . Hope it helps

Select the next item in the sequence.
10.172,10.983,10.994...

A. 10.972
B. 11.000
C.11.172
D.11.983

Answers

9514 1404 393

Answer:

  B. 11.000

Step-by-step explanation:

The function looks like a reflected and translated exponential function with a horizontal asymptote near y = 11.000. The rate of change is decreasing so fast that the next value is expected to be very near 10.994. The closest one among the answer choices is 11.000.

_____

First differences are 0.811 and 0.011. The latter is about 0.0136 times the former. At that rate of change, we expect the next first difference to be about 0.000149, which would make the next number in sequence be about 10.9941—very little change from 10.994.

Clearly, first differences are not constant, so the function is not linear. Ratios of the numbers are not constant, so this is not an exponential (geometric) sequence. A reflected exponential function of the type described is a good fit.

With only 3 points given, the rule is not at all obvious. The next term could legitimately be anything you like, and a rule could be made that would fit it.

How many different ways can you represent the weight in 1000g?​

Answers

8 different ways because there are 8 units for measuring weight.

A decreasing inventory turnover ratio indicates:_______
a. a longer time span between the purchase and sale of inventory.
b. a shorter time span between the purchase and sale of inventory.
c. a longer time span between the ordering and receiving of inventory.
d. a shorter time span between the ordering and receiving of inventory.

Answers

you’re answer is C. a longer time span between the ordering and receiving of inventory

What is the range of the function f(x) = 3x + 2 overthe interval -6 < x < 5?

Answers

Answer:

(-16, 17)

Step-by-step explanation:

The range of the function will be (-16, 17)

The range of function f(x) = 3x + 2 overthe interval -6 < x < 5 is (-13,14)

What is Range?

The difference between the highest and lowest values.

The given function is  f(x) = 3x + 2

f(x) equal tp three times of x plus two.

The given interval is minus six less than x less than five.

-6 < x < 5

Which means the x values are -5, -4, -3, -2, -1, 0, 1, 2,3,4

Plus in the function

f(-5)=-15+2=-13

f(-4)=-10

f(-3)=-9+2=-7

f(-2)=-6+2=-4

f(-1)=-3+2=-1

f(0)=0+2=2

f(1)=3+2=5

f(2)=6+2=8

f(3)=9+2=11

f(4)=12+2=14

Hence the range of function f(x) = 3x + 2 overthe interval -6 < x < 5 is (-13,14)

To learn more on Range click:

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In the figure above, AABC is an equilateral
triangle and each circle is tangent to the other
two circles. If each circle has diameter 10, what
is the distance h?
(A) 103
(B) 1513
(C) 15+513
(D) 10+1013
(E) 10+5/5

Answers

Answer:

B

Step-by-step explanation:

Women athletes at a certain university have a long-term graduation rate of 67%. Over the past several years, a random sample of 36 women athletes at the school showed that 23 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the university is now less than 67%? Use a 5% level of significance.
(a) What is the level of significance?
State the null and alternate hypotheses.
A. H0: p = 0.67; H1: p < 0.67
B. H0: p = 0.67; H1: p ≠ 0.67
C. H0: p < 0.67; H1: p = 0.67
D. H0: p = 0.67; H1: p > 0.67
(b) What sampling distribution will you use?
A. The standard normal, since np > 5 and nq > 5.
B. The standard normal, since np < 5 and nq < 5.
C. The Student's t, since np > 5 and nq > 5.
D. The Student's t, since np < 5 and nq < 5.
What is the value of the sample test statistic?
(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)
Sketch the sampling distribution and show the area corresponding to the P-value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
A. At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
B. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
C. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
D. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
(e) Interpret your conclusion in the context of the application.
A. There is sufficient evidence at the 0.05 level to conclude that the true proportion of women athletes who graduate is less than 0.67.
B. There is insufficient evidence at the 0.05 level to conclude that the true proportion of women athletes who graduate is less than 0.67.

Answers

Answer:

A. H0: p = 0.67; H1: p < 0.67

A. The standard normal, since np > 5 and nq > 5.;

Test statistic = - 0.397 ;

Pvalue = 0.3457;

D. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

B. There is insufficient evidence at the 0.05 level to conclude that the true proportion of women athletes who graduate is less than 0.67.

Step-by-step explanation:

p = 0.67 ; q = 1 - p = 1 - 0.67

Sample size, n = 36

x = 23

Test for normality :

(36*0.67) = 24.12

(36 * (1-0.67)) = 11.88

For a normal distribution :

np ≥ 5 and n(1 - p) ≥ 5

The hypothesis :

H0 : p = 0.67

H1 : p < 0.67

The Test statistic :

Z = (phat - p) / √[(p(1 -p))/n]

Phat = x / n = 23 / 36

Z = ((24/36) - 0.67)) / √[(0.67(1 -0.67))/36]

Z = - 0.031 / 0.0783687

Z = - 0.396983

Z = - 0.397

Usong the Pvalue from Z calculator ;

Pvalue of Z = 0.3457

If Pvalue < α ; Reject H0 ; If otherwise, Fail to reject H0

A campus deli serves 101 customers over its busy lunch period from 11:30 am to 1:30 pm. A quick count of the number of customers waiting in line and being served by the sandwich makers shows that an average of 18 customers are in process at any point in time. What is the average amount of time (in minute) that a customer spends in process?

Answers

Answer:

21.38 minutes

Step-by-step explanation:

the total number served = 101

teh number of hours that is spent in serving from 11.30 to 1.30 = 2 hours

Lq = average number of those that are in line = 18 customers

from here we are to find the flow rate

= [tex]\frac{total served}{service hour}[/tex]

= 101/2 = 50.5

the average time in waiting =

Lq/flow rate

= 18/50.5

= 0.3564 hours

convert to minutes = 0.3564 * 60 minutes

= 21.38 minutes

The tires Mary wants to buy for her car cost $100 per tire. A store is offering the following deal. Buy a tire and get the 4th tire for 75% off! Mary will buy 4 tires using the deal. A sales tax of 8% will be charged after applying the discount. How much money will Mary saveby using the deal instead of paying the full price for all 4 tires?

Answers

she saved $162

hope it helps


Help.. what does Y=

Answers

Answer:

The answer is B. 3

Step-by-step explanation:

The right angle points at side C which is the biggest side of a triangle and the smallest side is half of what side C is which is 3

Answer: B.3

Explanation: calculator.net, triangle calculator !

What is the answer to 6,1=? 8,2=20, 12,3=20, 10,5=52, 6,1=?

Answers

can u plz check if the question is right

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