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Xelga [282]
3 years ago
10

Let f(x) = 2x; its inverse is f1(x) = log2x. If f(5) = 32, what is f-1(32)?

Mathematics
1 answer:
PtichkaEL [24]3 years ago
3 0

Answer:

?

Step-by-step explanation:

You might be interested in
The plane x + y + z = 12 intersects paraboloid z = x^2 + y^2 in an ellipse.(a) Find the highest and the lowest points on the ell
emmasim [6.3K]

Answer:

a)

Highest (-3,-3)

Lowest (2,2)

b)

Farthest (-3,-3)

Closest (2,2)

Step-by-step explanation:

To solve this problem we will be using Lagrange multipliers.

a)

Let us find out first the restriction, which is the projection of the intersection on the XY-plane.

From x+y+z=12 we get z=12-x-y and replace this in the equation of the paraboloid:

\bf 12-x-y=x^2+y^2\Rightarrow x^2+y^2+x+y=12

completing the squares:

\bf x^2+y^2+x+y=12\Rightarrow (x+1/2)^2-1/4+(y+1/2)^2-1/4=12\Rightarrow\\\\\Rightarrow (x+1/2)^2+(y+1/2)^2=12+1/2\Rightarrow (x+1/2)^2+(y+1/2)^2=25/2

and we want the maximum and minimum of the paraboloid when (x,y) varies on the circumference we just found. That is, we want the maximum and minimum of  

\bf f(x,y)=x^2+y^2

subject to the constraint

\bf g(x,y)=(x+1/2)^2+(y+1/2)^2-25/2=0

Now we have

\bf \nabla f=(\displaystyle\frac{\partial f}{\partial x},\displaystyle\frac{\partial f}{\partial y})=(2x,2y)\\\\\nabla g=(\displaystyle\frac{\partial g}{\partial x},\displaystyle\frac{\partial g}{\partial y})=(2x+1,2y+1)

Let \bf \lambda be the Lagrange multiplier.

The maximum and minimum must occur at points where

\bf \nabla f=\lambda\nabla g

that is,

\bf (2x,2y)=\lambda(2x+1,2y+1)\Rightarrow 2x=\lambda (2x+1)\;,2y=\lambda (2y+1)

we can assume (x,y)≠ (-1/2, -1/2) since that point is not in the restriction, so

\bf \lambda=\displaystyle\frac{2x}{(2x+1)} \;,\lambda=\displaystyle\frac{2y}{(2y+1)}\Rightarrow \displaystyle\frac{2x}{(2x+1)}=\displaystyle\frac{2y}{(2y+1)}\Rightarrow\\\\\Rightarrow 2x(2y+1)=2y(2x+1)\Rightarrow 4xy+2x=4xy+2y\Rightarrow\\\\\Rightarrow x=y

Replacing in the constraint

\bf (x+1/2)^2+(x+1/2)^2-25/2=0\Rightarrow (x+1/2)^2=25/4\Rightarrow\\\\\Rightarrow |x+1/2|=5/2

from this we get

<em>x=-1/2 + 5/2 = 2 or x = -1/2 - 5/2 = -3 </em>

<em> </em>

and the candidates for maximum and minimum are (2,2) and (-3,-3).

Replacing these values in f, we see that

f(-3,-3) = 9+9 = 18 is the maximum and

f(2,2) = 4+4 = 8 is the minimum

b)

Since the square of the distance from any given point (x,y) on the paraboloid to (0,0) is f(x,y) itself, the maximum and minimum of the distance are reached at the points we just found.

We have then,

(-3,-3) is the farthest from the origin

(2,2) is the closest to the origin.

3 0
3 years ago
Declan draws triangle WXY. He then constructs a perpendicular bisector from vertex W that intersects side XY at point Z. What ca
Ganezh [65]

Answer:

YZ = XZ

Step-by-step explanation:

Perpendicular Bisector:

A perpendicular bisector of a line segment 'l' is a line that is perpendicular to the line segment 'l' and cuts the line segment 'l' into two equal parts.

Given:

1. A triangle WXY.

2. A perpendicular bisector from vertex W that intersects XY at point Z.

Conclusion based on the drawing:

a. Z is the midpoint of the line segment XY because point Z lies on the perpendicular bisector of XY.

b. Hence, XZ = YZ.

5 0
3 years ago
Read 2 more answers
Write 0.735 as a simplified fraction
Arisa [49]

Answer:

73/5

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Add Polynomials<br> (3y + y^3– 5) + (4y– 4y + 2y^3+ 8)
nexus9112 [7]

Step-by-step explanation:

Please refer to the attachment

8 0
3 years ago
as the schools sign language interpreter kiran gets paid 35.50 for every parent-teacher conference that he attends. He also gets
yKpoI14uk [10]

<u>The interpreter attended </u><u>22 parent-teacher conference</u><u> and </u><u>5 school related assembly</u>

To solve this problem, we would write out two set of linear equations.

The data given on this problem are

  • parent-teacher conference = $35.50
  • school related assembly = $42
  • The total number of engagements = 27

Let x represent the number of parent-teacher conference

let y represent the number of school related conference

<h3>Equations</h3><h3>x+y = 27...equation (i)\\35.50(x)+42y=991...equation(ii)</h3>

From equation (i)

x+y  = 27\\x = 27-y...equation(iii)\\

Put equation(iii) into equation (ii)

35.50x+42y=991\\35.50(27-y)+42y=991\\958.50-35.50y+42y=991\\6.50y=991-958.50\\y=5

Put y = 5 into equation 1

x+y=27\\y = 5\\x+5=27\\x=27-5\\x=22

From the above calculations, he attended 22 parent-teacher conference and 5 school related assembly.

Learn more on linear equations here;

brainly.com/question/4074386

8 0
2 years ago
Read 2 more answers
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