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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).

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X is less than or equal to 7 and greater than 3
bezimeni [28]
X≤7
x>3

As a result, the answer will be 3 less than x and less than or equal 7. Hope it help!
3 0
3 years ago
A square has a side length of 36 feet.This square is dilated by a scale factor of 2/3 to create a new square .What is the side l
Vaselesa [24]

Answer:

24 ft

Step-by-step explanation:

36 x 2/3= 24

It is dilated by a scale of 2/3 which means it will be smaller.

8 0
3 years ago
Holly made 12 pair of earrings for each of her girlfriends. To wrap the pairs individually, she creates a simple cardboard gift
Mekhanik [1.2K]

Answer:

3280cm²

Step-by-step explanation:

We are told the gift box is folded into a Pyramid with a square base.

Hence, we are to find the Surface Area of a square pyramid

From the question, we are given:

Length of the base of the square = 10cm

Triangle

Side length = 10cm

Height = 8 2/3 cm

A square Pyramid has 5 faces: 4 triangles and 1 square.

Step 1

Find the Area of the square

Formula = Side length² this is because all the side of a square is equal.

Length of the base = 10cm

Area of the square = (10cm)²

= 100cm²

Step 2

Find the Area of the triangle. We are told the 4 triangles are equilateral, which means their side are equal.

Hence, for one triangle.

Area of a triangle = 1/2 × base × height

Base = side length = 10cm

Height = 8 2/3 = 8.6666666667cm

Area of a triangle = 1/2 × 10 × 8.6666666667

Area of a triangle = 43.333333333cm²

Since we have 4 triangles

We would have : 4 × 43.333333333cm²

= 173.33333333cm²

The Area of the 4 equilateral triangles =

173.33333333cm²

Step 3

The Surface area of the Square Pyramid

= Sum of the areas of all the faces.

= 100cm² + 173.33333333cm²

= 273.33333333cm²

Therefore, the minimum amount of cardboard needed to make one gift box = 273.33333333cm²

Step 4

The minimum amount of cardboard Holly needs to create 12 gift boxes to wrap each pair of earrings individually is calculated as: 12 × 273.33333333cm²

= 3280cm²

7 0
3 years ago
HELP RIGHT NOW PIZZZ = FASTEST ANSWER BRAINLIEST AND I WILL THANK U
Nuetrik [128]

Answer:

1) The slope of the function g(x) is 0 and the slope of the function f(x) is -1.

2) The negative slope of the function f(x) shows that it is the line is increasing and the slope 0  of the function  g(x)  shows that the line will always have the same y-coordinate.

3) The slope of the function is f(x) is greater than the slope of the function  g(x).

Step-by-step explanation:

 For this exercise you need to know that the slope of any horizontal line is zero (m=0)

The slope of a line can be found with the following formula:

m=\frac{y_2-y_1}{x_2-x_1}

You can observe in the graph of the function  g(x)  given in the exercise, that this is an horizontal line.  Then,  you can conclude that its slope is:

m=0

The steps to find the slope of the function f(x) shown in the table attached, are the following:

- Choose two points, from the table:

(0,3) and (4,-1)

- You can say that:

y_2=-1\\y_1=3\\\\x_2=4\\x_1=0

- Substitute values into the formula m=\frac{y_2-y_1}{x_2-x_1}:

m=\frac{-1-3}{4-0}

- Finally, evaluating, you get:

m=\frac{-4}{4}\\\\m=-1

Therefore:

1) The slope of the function g(x) is 0 and the slope of the function f(x) is -1.

2) The negative slope of the function f(x) shows that it is the line is increasing and the slope 0  of the function  g(x)  shows that the line will always have the same y-coordinate.

3) The slope of the function is f(x) is greater than the slope of the function  g(x).

8 0
3 years ago
Do the quadrilaterals represent a dilation? why or why not?
Flura [38]

Answer: Yes it does represent a dilation... It is enlarged..


Step-by-step explanation:


4 0
3 years ago
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