Please helppppppppppppp!!!!!!!!!!!!!!!!!!!!

Step-by-step explanation:

Content

Functions and their inverses

We begin with a simple example.

Example

Let f(x)=2x and g(x)=x2.

Apply the function g to the number 3, and then apply f to the result:

g(3)=32andf(32)=3.

A similar thing happens if we first apply f and then apply g:

f(3)=6andg(6)=3.

It is clear that this will happen with any starting number. This is expressed as

f(g(x))g(f(x))=x,for all x=x,for all x.

The function f reverses the effect of g, and the function g reverses the effect of f. We say that f and g are inverses of each other.

As another example, we have

(x−−√3)3=xandx3−−√3=x,

for all real x. So the functions f(x)=x3 and g(x)=x−−√3 are inverses of each other.

If x≥0, then (x−−√)2=x and x2−−√=x. If x<0, then x−−√ is not defined. So the functions f(x)=x2 and g(x)=x−−√ are inverses of each other, but we need to be careful about domains. We will look at this more carefully later in this section.

Basics

In an earlier section of this module, we defined the composite of two functions h and g by (g∘h)(x)=g(h(x)).

Definitions

The zero function 0–:R→R is defined by 0–(x)=0, for all x.

The identity function id:R→R is defined by id(x)=x, for all x.

Example

Consider a function f:R→R.

Prove that

0–∘f=0–

f∘id=f

id∘f=f.

Show that f∘0– does not necessarily equal 0–.

Solution

We have (0–∘f)(x)=0–(f(x))=0, for all x, and so 0–∘f=0–.

We have (f∘id)(x)=f(id(x))=f(x), for all x, and so f∘id=f.

We have (id∘f)(x)=id(f(x))=f(x), for all x, and so id∘f=f.

Consider the function given by f(x)=2, for all x. Then f∘0–(x)=f(0–(x))=f(0)=2, and so f∘0–≠0–.

Definition

Let f be a function with both domain and range all real numbers. Then the function g is the inverse of f if

f(g(x))g(f(x))=x,for all x,and=x,for all x.

That is, f∘g=id and g∘f=id.

Notes.

Clearly, if g is the inverse of f, then f is the inverse of g.

We denote the inverse of f by f−1. We read f−1 as 'f inverse'. Note that f inverse has nothing to do with the function 1f.

Example

Let f(x)=x+2 and let g(x)=x−2. Show that f and g are inverses of each other.

Solution

We have

f(g(x))=f(x−2)=x−2+2=x,for all x(f∘g=id)

and

g(f(x))=g(x+2)=x+2−2=x,for all x(g∘f=id).

Hence, the functions f and g are inverses of each other.

Exercise 5

Find the inverse of

f(x)=x+7

f(x)=4x+5.

Example

Let f(x)=ax+b with a≠0. Find the inverse of f.

Solution

We have x=f(x)−ba, for all x. So let g(x)=x−ba. Then

f(g(x))g(f(x))=f(x−ba)=a(x−ba)+b=x=g(ax+b)=(ax+b)−ba=x,

for all x. Hence, g is the inverse of f.

Exercise 6

Show that f(x)=x5 and g(x)=x15 are inverses of each other.

Find the inverse of f(x)=x3+2.

We do not yet have a general enough concept of inverses, since x2 and x−−√ do not fit into this framework, nor do ex and logex. We will give a definition that covers these functions later in this section.

The horizontal-line test

Consider the function f(x)=x2, which has domain the reals and range A={x:x≥0}. Does f have an inverse?

The following graph shows that it does not. We have f(−2)=f(2)=4, and so f−1(4) would have to take two values, −2 and 2! Hence, f does not have an inverse.

Graph of y = x squared and the line y = 4 on the one set of axes.

This idea can be formulated as a test.

Horizontal-line test

Let f be a function. If there is a horizontal line y=c that meets the graph y=f(x) at more than one point, then f does not have an inverse.

Notes. Remember that the vertical-line test determines whether a relation is a function.

Example

Consider the function

f(x)=x3−x=(x+1)x(x−1).

Its graph is shown in the following diagram.

Graph of y = x cubed minus x.

Does f have an inverse?

Solution

The line y=0 meets the graph at three points. By the horizontal-line test, the function f does not have an inverse.

The function whose inverse does not exist is f(x) = x(x - 1)

The correct option is (D)  f(x) = x(x - 1)

An inverse is a function that serves to “undo” another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x.

First,  f(x)= [tex]\frac{1}{2} x -\frac{1}{2}[/tex]

let y= [tex]\frac{1}{2} x -\frac{1}{2}[/tex]

On solving for x we get a unique value

Then replace x and y.

It shows that the function have a unique value, which satisfies the condition of inverse.

Now, f(x) =[tex](x - 1)^3 + 2[/tex]

Again, solving for y we can get a cube root function which is a inverse of cube.

Hence, the inverse of [tex](x - 1)^3 + 2[/tex] exists.

Next, f(x) =[tex]2^x[/tex]

Solving for above we get logarithmic value. Log function are inverse of exponential function.

Hence, the inverse of [tex]2^x[/tex] exists.

Last, f(x)= x(x-1)

Solving for above create a square value.

The inverse of square never exist because having square root gives two value one is positive and other is negative.

Hence, the inverse of x(x-1) not exists.

Hence the function whose inverse does not exist is x(x-1).

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Answer:

the answer is 1.6

Step-by-step explanation:

Answer:i have no Step-by-step explanation:

a

Answer:

27 cm

Step-by-step explanation:

the radious of the pot:

[tex]v=\pi r^{2} h[/tex]

[tex]264=\pi r^{2} (8)[/tex]

[tex]r^{2} =\frac{264}{8\pi }[/tex]

The height of pot B

[tex]891=\pi (\frac{264}{8\pi })h[/tex]

[tex]891=(\frac{264}{8} )h[/tex]

[tex]h=\frac{(891)(8)}{264} =27[/tex]

I hope this help you

Answer:

when you have the two points of a line (-4,5) and (0,6) then you use the m = (y2-y1)/(x2-x1) and you get m = 1/4

Answer:

(3x)¹²

Step-by-step explanation:

(3x³)⁴ can be simplified into (3x)¹²

The simplified value of the given expression is 3x¹²

Simplification is the process of replacing a mathematical expression by an equivalent one, that is simpler, usually shorter.

Given that, an expression (3x³)⁴

Using the identity = (xᵃ)ᵇ = xᵃᵇ

(3x³)⁴ = (3x)³ˣ⁴ = 3x¹²

Hence, the simplified value of the given expression is 3x¹²

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The missing factor that makes the equation true is 0.2.

According to the question;

Let the missing factor be represented as x.

In essence; by dividing both sides of the equation by 1.1; we have;

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Answer:

Triangle A) acute

Triangle B) right

Triangle C) obtuse

Triangle D) obtuse

Step-by-step explanation:

Any triangle less than 90° is acute.

Triangles at exactly 90° are right triangles.

Any triangle more than 90° are called obtuse.

Simplify the expression

90 + 4B - 2B - 7B

Already in like terms, therefore

90+2B-7B

90-5B

Please Mark BRAINLIEST.

Answer:

Worked 5 hours: $550

Worked 7 hours: $2,750

Worked 12 hours: $6,600

Step-by-step explanation:

5+7+12=24

13200/24 = 550

(5)*(550) = 2,750

(7)*(550) = 3,850

(12)*(550) = 6,600

Answer:

How

Step-by-step explanation:

You want to put picture in

Answer:

The inverse matrix is used to solve the system of linear equations. It also tells us the consistent or inconsistent behaviour of the solution of equations.

Step-by-step explanation:

In real life,

Answer:

Option 2: A=1, B=3, C=4

Step-by-step explanation:

Answer:

3, 9, 81, 6561

Step-by-step explanation:

previous term × itself

[tex]\\ \sf\longmapsto f(n)=[f(n-1)]^2[/tex]

[tex]\\ \sf\longmapsto f(2)=(f(1))^2=3^2=9[/tex]

[tex]\\ \sf\longmapsto f(3)=(f(2))^2=(9)^2=81[/tex]

[tex]\\ \sf\longmapsto f(4)=(f(3))^2=(81)^2=6561[/tex]

Answer:

y=6x+10a

Step-by-step explanation:

let the number of children be x

and the number of adults be a

let the total price be y

given that children tickets are $6 each

also, an adult ticket is $10

x+a=14-------1

y=6x+10a----2

Therefore the expression for the total price can be expressed as

 y=6x+10a

We are required to find the price of the first game

The price of the first game is $20

Given:

let

price of the first game = x

price of the second game = x + 50% of x

= x + 0.5x

= 1.5x

Price of the third game = 50% of 1.5x

= 0.5 × 1.5x

= 0.75x

Total cost of game = $65

Total cost of game = price of the first game + price of the second game + Price of the third game

65 = x + 1.5x + 0.75x

65 = 3.25x

divide both sides by 3.25

x = 65 ÷ 3.25

x = 20

Therefore, the price of the first game is $20

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Answer:

y=4/5x-0.8

Step-by-step explanation:

Answer:[tex]x 6[/tex]

Step-by-step explanation: Im not to good at math but hope that hlps

Answer:

x^x

Step-by-step explanation:

hope this helps :)

Answer:

6.3

Step-by-step explanation:

1) rule:

[tex]d=\sqrt{(X_A-X_B)^2+(Y_A-Y_B)^2)}[/tex]

2) according to the rule above:

[tex]d=\sqrt{36+4}=2\sqrt{10};[/tex]

d≈6.3

Answer:

6.3

Step-by-step explanation:

Answer:

-28

Step-by-step explanation:

You have to do the parenthesis 1st, then the [tex]\frac{8}{3}[/tex], and multiply the result :D

Answer:

0.9m²

Step-by-step explanation:

Find the area of the larger circle, the radius is 0.85m because the diameter is double this - 1.7m:

Find the area of the smaller circle, the radius is 0.65 because the diameter is double this - 1.3m:

Subtract the two area:

289/400π - 169/400π = 120/400π = 3/10π or 0.942477.. ≈ 0.9m²

Answer:

Step-by-step explanation:

Answer:

1)

Angle 1: 59

Angle 2: 88

Angle 3: 147

2)x =21

Step-by-step explanation:

1)

Angle on a straight line add up to 180 degrees and angles in a triangle add to 180 degrees

2)

Opposite angles are equal. (4x) and (2x+42) are opposite therefore equal, so

4x=2x+42

Minus 2x from both sides

2x = 42

Find 1x by dividing 42 by 2, therefore

X=21

Step-by-step explanation:

Forced vital capacity: the maximum amount of air you can forcibly exhale from your lungs after fully inhaling. It is about 80 percent of total capacity, or 4.8 liters, because some air remains in your lungs after you exhale

9514 1404 393

Answer:

  a) 1.70 L

  b) 16 breaths per minute

Step-by-step explanation:

a) The maximum value of v(t) occurs when the sine function is at its maximum value of 1. In that case, the volume is ...

  1.27 +0.43×1 = 1.70 . . . liters

__

b) One full breath cycle takes a period of time that makes the argument of the sine function be 2π. The that time period is ...

  1.71t = 2π

  t = 2π/1.71 ≈ 3.6744 . . . seconds

Then the number of breaths per minute is ...

  (60 seconds/minute)/(3.6744 seconds/breath) ≈ 16.33 breaths/minute

The person takes about 16 breaths per minute while sleeping.

Step-by-step explanation:

5x-7+3x+27=180

8x+20=180

8x=160

x=20°.

hope this helps you.

Answer:

[tex]5x - 7 + 3x + 27 = 180 \\ 5x + 3x + 27 - 7 = 180 \\ 8x + 20 = 180 \\ 8x = 180 - 20 \\ 8x = 160 \\ d \: b \: s \: by \: 8 \\ 8x \div 8 = 160 \div 8 \\ x = 20[/tex]

Answer:

(

2

,

8

)

Explanation:

Two lines are parallel if they share the same slope.

Any line parallel to

y

=

10

x

will have the slope 10.

Now, we have:

y

=

3

x

2

−

2

x

We try to find

d

y

d

x

=>

d

d

x

(

y

)

=

d

d

x

(

3

x

2

−

2

x

)

Step-by-step explanation:

Answer:

27,28 and 29

Step-by-step explanation:

10*a = a+1+a+2+213

10a - 2a = 216

8a = 216

a = 27

27, 28, and 29

Answer:

0.521

Step-by-step explanation:

It's just like subtracting