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
Liono4ka [1.6K]
3 years ago
13

If u(x)=-2x²+3 and v(x)=1/x, what is the range of (u ° v)(x)?

Mathematics
2 answers:
Ludmilka [50]3 years ago
6 0

u(x)=-2x^2+3\\\\v(x)=\dfrac{1}{x}\\\\(u\ \circ\ v)(x)=-2\left(\dfrac{1}{x}\right)^2+3=-2\left(\dfrac{1}{x^2}\right)+3=-\dfrac{1}{x^2}+3\\\\\text{The range of}\ y=\dfrac{1}{x^2}\ \text{is all positive real numbers.}\\\\\text{The range of}\ y=-\dfrac{1}{x^2}\ \text{is all negative real numbers.}\\\\\text{The range of }\ (u\ \circ\ v)(x)=-\dfrac{1}{x^2}+3\ is\ (-\infty,\ 3)

Anna11 [10]3 years ago
4 0

Answer:

(-∞,3)

Step-by-step explanation:

Imagine functions as little machines that turn one number into another number, the set of numbers that can enter the machine are called domain and the set of  numbers that can exit the machine are called range.  In this excercise we have to do function composition, which would be like having two machines and take the numbers that exit machine number 2 (v(x)) and enter them into the machine number 1 (u(x)). In math this is done by replacing the x in u(x) with the function v(x) like this:

u(x)=-2x^{2} +3\\v(x)=\frac{1}{x}\\(u°v)=-2(\frac{2}{x})^{2} +3

Now we need the range of the composed function, this will be the numbers that can come out of the machine number 1 when the numbers from the machine number 2 are entered to it.

So first which numbers will come out of machine number 2(v(x))? All but 0 because  we there is no number that we can divide 1 by it that will give us the value 0 (not even zero itself because it is and indeterminate form).

We have now which numbers will enter machine number 1 (we dont have any restrictions in u(x) to enter numbers)

The range of the composed function will be then the range of u(x) less the value that we woud obtain by replacing x with 0.

The range of u(x) is (-∞,3] according to the  attached graph and the value that we woud obtain by replacing x with 0 is 3 so we would have (-∞,3).

You might be interested in
What is the result when the number 40 is increased by 10%?
Goshia [24]

The answer is 44.

40 / 100 = 0,4 * 10 = 4 (10%)

40 + 4 = 44

7 0
3 years ago
Read 2 more answers
Y+5=2(x+1)<br><br><br> what is the slope for this Math
Stells [14]

Answer:

2

Step-by-step explanation:

The slope-intercept form is

y=mx+b, where m is the slope and b is the y-intercept.y=mx+bRewrite in slope-intercept form.Simplify 2(x+1).Apply the distributive property.

y+5=2x+2⋅1

multiply 2 by 1.

y+5=2x+2

y+5=2x+2

Move all terms not containing y to the right side of the equation.Subtract 5 from both sides of the equation.

y=2x+2−5

y=2x−3

the slope is the "m" so the slope is 2

7 0
3 years ago
a rectangle is 14 inches long and 4 inches wide. a smaller, similar rectangle is 2 inches wide. to the nearest inch what is the
KonstantinChe [14]

Answer:

7

Step-by-step explanation:

4/2 = 14/x, cross multiply and simplify, x=7

5 0
3 years ago
Read 2 more answers
What rational number equals 0.1111111111
Katen [24]

Answer:

1/9

Step-by-step explanation:

1/9 is a fraction, so it is rational.

6 0
2 years ago
Read 2 more answers
39-50 find the limit.<br> 41. <img src="https://tex.z-dn.net/?f=%5Clim%20_%7Bt%20%5Crightarrow%200%7D%20%5Cfrac%7B%5Ctan%206%20t
Katyanochek1 [597]

Write tan in terms of sin and cos.

\displaystyle \lim_{t\to0}\frac{\tan(6t)}{\sin(2t)} = \lim_{t\to0}\frac{\sin(6t)}{\sin(2t)\cos(6t)}

Recall that

\displaystyle \lim_{x\to0}\frac{\sin(x)}x = 1

Rewrite and expand the given limand as the product

\displaystyle \lim_{t\to0}\frac{\sin(6t)}{\sin(2t)\cos(6t)} = \lim_{t\to0} \frac{\sin(6t)}{6t} \times \frac{2t}{\sin(2t)} \times \frac{6t}{2t\cos(6t)} \\\\ = \left(\lim_{t\to0} \frac{\sin(6t)}{6t}\right) \times \left(\lim_{t\to0}\frac{2t}{\sin(2t)}\right) \times \left(\lim_{t\to0}\frac{3}{\cos(6t)}\right)

Then using the known limit above, it follows that

\displaystyle \left(\lim_{t\to0} \frac{\sin(6t)}{6t}\right) \times \left(\lim_{t\to0}\frac{2t}{\sin(2t)}\right) \times \left(\lim_{t\to0}\frac{3}{\cos(6t)}\right) = 1 \times 1 \times \frac3{\cos(0)} = \boxed{3}

4 0
1 year ago
Other questions:
  • Write the standard form of the line that passes through the given points.
    8·1 answer
  • the radius of a cylindrical gift box is (4x+3) inches. The height of the gift box is three times the radius. what is the surface
    6·1 answer
  • An alloy of tin is 14% tin and weighs 22 pounds. A second alloy is 9% tin. How much of the second alloy must be added to the fir
    5·2 answers
  • What type of triangle is this?​
    6·2 answers
  • A number that is at most 13
    15·2 answers
  • What is scale factor
    14·2 answers
  • Draw a sketch to help you solve this problem.
    6·2 answers
  • Which of the following has the larger area?
    10·2 answers
  • Manuel bought 9 pounds of apples.
    5·1 answer
  • Steve reaches into a bag with 5 marbles 3 yellow marbles and 2 green marbles and randomly selects one the he puts the marble bac
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!