1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
lisabon 2012 [21]
3 years ago
14

Let f be the function given by f(x)= (x-1)(x^2-4)/x^2-a. For what positive values of a is f continuous for all real numbers x?

Mathematics
1 answer:
9966 [12]3 years ago
4 0

Answer:

a =1 and a=4.

Step-by-step explanation:

The function is

f(x)=\frac{(x-1)(x^2-4)}{x^2-a}

If we want f(x) to be continuous the denominator needs to be different to 0, otherwise f(x) will be indeterminate.

Now, for a a positive real we have that x=\sqrt{a} will annulate the denominator, i.e

(\sqrt{a})^2-a = a-a = 0. But, if a = 1 we have:

f(x)=\frac{(x-1)(x^2-4)}{x^2-1} = \frac{(x-1)(x^2-4)}{(x-1)(x+1)}=\frac{(x^2-4)}{x+1}

so, the value x=\sqrt{a} = \sqrt{1} = 1 won't annulate the denominator.

Now, for a = 4 we have:

f(x)=\frac{(x-1)(x^2-4)}{x^2-4} = x-1

so, the value x=\sqrt{a} = \sqrt{4} = 2 won't annulate the denominator.

In conclusion, for a=1 or a=1, the function will be continuos for all real numbers, since the denominator will never be 0.

You might be interested in
javon is helping his dad build a tree house. he has a piece of trim that is 13 ft long. how many pieces can Javon cut that are 1
wel
Well a yard is 3 ft
so  divide 3 into 13
3 goes into 13 4 times and you get 12
you have 4 whole yards and 1 ft left or 1/3 of a yard.
:) hoped this helped

6 0
3 years ago
Read 2 more answers
The owner of a small pet supply store wants to open a second store in another city, but he only wants to do so if more than one-
Andrei [34K]

Answer:

The answer is "The null hypothesis was rejected".

Step-by-step explanation:

Following are the right-tailed test:

Calculating the null and alternative hypothesis:

H_0 : p = 0.3333\\\\H_a : p > 0.3333\\\\n = 150\\\\x = 64\\\\\hat{p} = \frac{x}{n} = \frac{64}{150} = 0.4267\\\\P_0 = 0.3333\\\\1 - P_0 = 1 - 0.3333 = 0.6667\\\\

z = \frac{\hat{p} - P_0}{[\sqrt{\frac{P_0 \times (1 - P_0 )}{n}}]}

   = \frac{0.4267 - 0.3333}{[\sqrt{\frac{(0.3333 \times 0.6667)}{150}}]}\\\\= 2.426

Calculating the right-tailed test:

P(z > 2.426) = 1 - P(z < 2.426) = 1 - 0.9924 = 0.0076\\\\P-value = 0.0076\\\\\alpha = 0.05\\\\P-value < \alpha

Therefore, we reject the null hypothesis.

This example shows that more than a third of the families own pets in this town.

5 0
2 years ago
Find the vertex of the graph of the function.
lapo4ka [179]

Answer: The correct options are  (1) (5,10), (2) (3,-3), (3) x = -1, (4) y=(x+2)^2+3, (5) 21s and (6) 0, -1, and 5.

Explanation:

Te standard form of the parabola is,

f(x)=a(x-h)^2+k        .....(1)

Where,  (h,k) is the vertex of the parabola.

(1)

The given equation is,

f(x)=(x-5)^2+10

Comparing this equation with equation (1),we get,

h=5 and k=10

Therefore, the vertex of the graph is (5,10) and the fourth option is correct.

(2)

The given equation is,

f(x)=3x^2-18x+24

f(x)=3(x^2-6x)+24

To make perfect square add (\frac{b}{2a})^2, i.e., 9. Since there is factor 3 outside the parentheses, so subtract three times of 9.

f(x)=3(x^2-6x+9)+24-3\times 9

f(x)=3(x-3)^2-3

Comparing this equation with equation (1),we get,

h=3 and k=-3

Therefore, the vertex of the graph is (3,-3) and the fourth option is correct.

(3)

The given equation is

f(x)=4x^2+8x+7

f(x)=4(x^2+2x)+7

To make perfect square add (\frac{b}{2a})^2, i.e., 1. Since there is factor 4 outside the parentheses, so subtract three times of 1.

f(x)=4(x^2+2x+1)+7-4

f(x)=4(x+1)^2+3

Comparing this equation with equation (1),we get,

h=-1 and k=3

The vertex of the equation is (-1,3) so the axis is x=-1 and the correct option is 2.

(4)

The given equation is,

y=x^2+4x+7

To make perfect square add (\frac{b}{2a})^2, i.e., 2^2.

f(x)=x^2+4x+4+7-4

f(x)=x^2+4x+4+7-4

f(x)=(x+2)^2+3

Therefore, the correct option is  4.

(5)

The given equation is,

h=-16t^2+672t

It can be written as,

h=-16(t^2-42t)

It is a downward parabola. so the maximum height of the function is on its vertex.

The x coordinate of the vertex is,

x=\frac{b}{2a}

x=\frac{42}{2}

x=21

Therefore,  after 21 seconds the projectile reach its maximum height and the correct option is first.

(6)

The given equation is,

f(x)=3x^3-12x^2-15x

f(x)=3x(x^2-4x-5)

Use factoring method to find the factors of the equation.

f(x)=3x(x^2-5x+x-5)

f(x)=3x(x(x-5)+1(x-5))

f(x)=3x(x-5)(x+1)

Equate each factor equal to 0.

x=0,-1,5

Therefore, the zeros of the given equation is 0, -1, 5 and the correct option is 2.

3 0
3 years ago
Let p: The integer has one digit. Let q: The integer is less than 10. Which represents "The integer has one digit if and only if
Vinil7 [7]

The double implication "if and only if" is graphically represented by an arrow pointing in both direction, so the correct writing is

p \iff q

8 0
3 years ago
A tire travels about 75 inches in one full rotation. What is the Demeter of the tire, to the nearest inch?
Gnoma [55]

Answer:

<h2><u>24 inches</u></h2><h2><u></u></h2>

Step-by-step explanation:

c = πd

75 = 3.14d

Divide both sides by 3.14

23.89 = d

Rounded

24 inches

5 0
3 years ago
Other questions:
  • Help with my question :<br><br> 8x•6-4^3(8x•3)-45=12
    9·1 answer
  • If R is midpoint of QS and QS=40 find QR
    6·1 answer
  • Describe the relationship between the two quantities. Distance traveled by car; Amount of gas in the car.
    11·2 answers
  • END OF GRADING CYCLE AND SO MANY MISSING ASSIGNMENTS PLEASE HELP
    13·1 answer
  • Solve the proportion <br> 7x + 1 9 = 2x 3
    10·1 answer
  • Find the measure of Angle F ASAP thank u !!
    8·2 answers
  • A market sells blueberries for $3 per pint. The first day, the market sold a total of $35 worth of blueberries. The second day,
    12·1 answer
  • Which answer is correct
    11·1 answer
  • Guys need help with this question NOTE: the answer is a mixed number not a whole number.
    13·1 answer
  • Given f(x) = x3 - x2 + 4x - 1 and<br> g(x) = -2x, find lim g(f(x))<br> X-2
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!