Identify the rate of change of the function.

Answer Before defining the inverse of a function we need to have the right mental image of function.

Consider the function f(x) = 2x + 1. We know how to evaluate f at 3, f(3) = 2*3 + 1 = 7. In this section it helps to think of f as transforming a 3 into a 7, and f transforms a 5 into an 11, etc.

Now that we think of f as "acting on" numbers and transforming them, we can define the inverse of f as the function that "undoes" what f did. In other words, the inverse of f needs to take 7 back to 3, and take -3 back to -2, etc.

Let g(x) = (x - 1)/2. Then g(7) = 3, g(-3) = -2, and g(11) = 5, so g seems to be undoing what f did, at least for these three values. To prove that g is the inverse of f we must show that this is true for any value of x in the domain of f. In other words, g must take f(x) back to x for all values of x in the domain of f. So, g(f(x)) = x must hold for all x in the domain of f. The way to check this condition is to see that the formula for g(f(x)) simplifies to x.

g(f(x)) = g(2x + 1) = (2x + 1 -1)/2 = 2x/2 = x.

This simplification shows that if we choose any number and let f act it, then applying g to the result recovers our original number. We also need to see that this process works in reverse, or that f also undoes what g does.

f(g(x)) = f((x - 1)/2) = 2(x - 1)/2 + 1 = x - 1 + 1 = x.

Letting f-1 denote the inverse of f, we have just shown that g = f-1.

Definition:

Let f and g be two functions. If

f(g(x)) = x and g(f(x)) = x,

then g is the inverse of f and f is the inverse of g.

Exercise 1:

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Finding Inverses

Example 1. First consider a simple example f(x) = 3x + 2.

The graph of f is a line with slope 3, so it passes the horizontal line test and does have an inverse.

There are two steps required to evaluate f at a number x. First we multiply x by 3, then we add 2.

Thinking of the inverse function as undoing what f did, we must undo these steps in reverse order.

The steps required to evaluate f-1 are to first undo the adding of 2 by subtracting 2. Then we undo multiplication by 3 by dividing by 3.

Therefore, f-1(x) = (x - 2)/3.

Steps for finding the inverse of a function f.

Replace f(x) by y in the equation describing the function.

Interchange x and y. In other words, replace every x by a y and vice versa.

Solve for y.

Replace y by f-1(x).

Example 2.  f(x) = 6 - x/2

Step 1 y = 6 - x/2.

Step 2 x = 6 - y/2.

Step 3 x = 6 - y/2.

y/2 = 6 - x.

y = 12 - 2x.

Step 4 f-1(x) = 12 - 2x.

Step 2 often confuses students. We could omit step 2, and solve for x instead of y, but then we would end up with a formula in y instead of x. The formula would be the same, but the variable would be different. To avoid this we simply interchange the roles of x and y before we solve.

Example 3.  f(x) = x3 + 2

This is the function we worked with in Exercise 1. From its graph (shown above) we see that it does have an inverse. (In fact, its inverse was given in Exercise 1.)

Step 1 y = x3 + 2.

Step 2 x = y3 + 2.

Step 3 x - 2 = y3.

(x - 2)^(1/3) = y.

Step 4 f-1(x) = (x - 2)^(1/3).

Exercise 3:

Graph f(x) = 1 - 2x3 to see that it does have an inverse. Find f-1(x). Answer

Step-by-step explanation:

pls brain list

Answer:

1. Point C

2. a clockwise

Step-by-step explanation:

Got the question right

Answer:

correct answer below

Step-by-step explanation:

Answer:

g(x) is f(x) stretched vertically by a factor of 4

Step-by-step explanation:

Since g(x) = 4 x^2, the graph of g(x) is a vertically stretched (by a factor of 4) version of the graph of f(x).

Answer:

kite

A quadrilateral is a kite if: one diagonal bisects the vertex angles through which it passes, or. one diagonal is the perpendicular bisector of the other diagonal.

Step-by-step explanation:

not %sure

Answer:

9/5

Step-by-step explanation:

I hope this helps im pretty sure its correct

Answer:

10%

Step-by-step explanation:

Answer:

Okay so the answer is not B. How many miles did i run today.

Step-by-step explanation:

Answer:

[tex]24ft^{2}[/tex]

Step-by-step explanation:

If you break the shapes into two sections you get

2 × 4 = 8 & 8 × 2 = 16

16 + 8 = 24

good luck, i hope this helps :)

Answer:

56766.97

Step-by-step explanation:

Answer:

46 ft

Step-by-step explanation:

A = π×d²/4

d² = √(4×A /π)

= √(4×1611.06 / 3.14)

=√ 2116

= 46 ft

Answer: 4 deeds per day

Step-by-step explanation: 7 times 4 is 28

Answer:

4. (d, the last one)

5. (a, the first one)

Step-by-step explanation:

Answer: I'm late but do you still need the answer?

Step-by-step explanation:

Answer:

Today: Friday, 12th Feb 2021

10.11 PM

_______________________

[tex] \rm 100 {x}^{2} \times 100{x}^{2} [/tex]

[tex] \rm = (100 \times 100)^{2 + 2} [/tex]

[tex] {\boxed{\boxed{ \rm = 10000x^{4} }}}[/tex]

-

[tex]{\boxed{\boxed{ \rm \pink{i} \: \blue{hope} \: \orange{this} \: \pink{help} \: \blue{you} \: : )}}}[/tex]

Answer:

100ײ×100x²=100×100×x²×x²=10000x⁴ is your answer

Answer:

x(3 + 9) = 3x + 9x = 12x

Step-by-step explanation:

...........

Answer:

1

Step-by-step explanation:

Answer:

4sides

Step-by-step explanation:

these is answer 4 sided

Answer:

es 3/54 eso da My brother

Answer:

I think its C

Step-by-step explanation:

Answer:

Depends on how many gallojs but 4 quarts per gallon

Step-by-step explanation:

Answer:

it is 24

Step-by-step explanation:

I just know it cuz I did it. hope it helps u :)

Answer:

25

Step-by-step explanation:

In this it's going to be pretty simple.

[tex]b^{2}[/tex] is going to be -3 * -3 = 9

When you have a square you multiply the number by itself.

So

9 - 4 * 1 * -4

here you have to remember order of operations.

Please - Parens

Excuse - Exponents

My - Multiplication

Dear - Division

Aunt - Addition

Sally - Subtraction.

Multiplication and Division are done at the same time left to right

Addition and Subtraction are done at the same time left to right.

So

9 - 4 * 1 * -4

You do the multiplication first.

4*1 = 4

4 * -4 = -16

Now you have

9 - - 16

This turns into

9 + 16

your answer is 25.

Answer:

78 pages

Step-by-step explanation:

Find out 22% using 1% of the number of photos the album can hold.

100% = 800

1% = 800 ÷ 100 = 8

22% = 8 x 22 = 176

800 - 176 = 624

The album has 624 photos.

If 8 photos can create a page...

624 ÷ 8 = 78 pages

Answer:

the answer is B

Answer:

or 1/50

Step-by-step explanation:

Q. What is the fractional form of 0.02?

→ 0.02 = 002/100 = 2/100

HOPE THIS HELPS YOU

Follw me = 10 thanks

Mark as brainliest = 10 thanks

1 thanks to me = 2 thanks

Answer:

1 1/8

8 goes into 9 one time, leftovers is 1. Keep the bottom number. 1 1/8

Answer:

11/3 is the answer

Step-by-step explanation:

devide it

Answer:

one and twenty-five and five

if u do one times twenty-five and then five times five

1, 5, 25

The factors are all the numbers that could be used to multiply together to equal 25. 1×25=25, and 5×5=25.

pesan 3kilos.

3/1=0.3kilos

Answer:

1 1/3 oz per day

Step-by-step explanation:

1/2 lb = 8 oz

8 oz/6 days = 1 1/3 oz per day