Question

math problem

Answer 1

@AZ

Answer 2

Similar to your previous question

To find the inverse, you have to

1) Change f(x) to y
2) Switch x and y
3) Solve for y
4) Replace y with \( f^{-1}(x)\)

Answer 3

im not sure

Answer 4

Let's do it one step at a time then

We have

\( f(x) = -9\sqrt{x-8}+5\)

Answer 5

The first thing we want to do is replace f(x) with y

so we get

\( y = -9 \sqrt{x-8}+5\)

Does that make sense so far?

Answer 6

yep

Answer 7

The second step says to switch x and y

So we go from
\( y = -9\sqrt{x-8}+5\)

and we switch the x and y

\( x= -9\sqrt{y-8}+5\)

Still following along? All we did is swapped the two letters

Answer 8

yep

Answer 9

Now we need to solve for y

Can you do that?

\( x = -9 \sqrt{y-8} + 5\)


First add 5 to both sides. What do you get?

Answer 10

-3

Answer 11

y-3

Answer 12

az

Answer 13

Oops I meant subtract 5 on both sides

What do you get?

\( x - 5 = -9 \sqrt{y-8} + 5 - 5\)

what is 5 - 5 =

Answer 14

0

Answer 15

Good so now we have

\( x - 5 = -9 \sqrt{y-8}\)

We want to get the square root all by itself, so what do you get if you divide both sides by -9?

Answer 16

9 divided by 8 or 8 divided by 0

Answer 17

9 divided by 8 or 8 divided by 0

Answer 18

8 divided by 9

Answer 19

No

We get

\( \dfrac{x-5}{-9} = \dfrac{-9\sqrt{y-8}}{-9}\)

Do you see how the -9 cancels out on the right side?

Answer 20

9 divided by 9?

Answer 21

-9 / - 9

Answer 22

its srt y-8

Answer 23

az?

Answer 24

yes

Answer 25

so now we have

\( \dfrac{x-5}{-9} = \sqrt{y-8}\)

How would we get rid of the square root?

Could we square both sides to get rid of it?

Answer 26

yep?

Answer 27

So what do you get when you do that?

Answer 28

y-8

Answer 29

Good! And on the other side we would get all that ^2

So we have

\( \left(\dfrac{x-5}{-9} \right) ^2 = y - 8\)

Answer 30

Do you want to simplify what we have on the left side first?

Answer 31

yep

Answer 32

az

Answer 33

az?

Answer 34

yae

Answer 35

So what is

(x-5)^2 =

Answer 36

Remember that

\(( a - b)^2 = a^2 - 2ab + b^2\)

Answer 37

x^2-10x+25

Answer 38

Good so now we have

\( \dfrac{x^2 - 10x + 25}{81} = y + 8\)

Subtract 8 on both sides now

Answer 39

y

Answer 40

yes

what about on the left side?

can you write 8 as maybe

8*81 / 81

and then subtract the number

Answer 41

8-8?

Answer 42

no

what is 8 * 81 =

Answer 43

648

Answer 44

good

so we have

\( \dfrac{x^2 - 10x + 25}{81} + 8 = y \)


Now we change that 8 so it has a denominator of 81

\( \dfrac{x^2 - 10x + 25}{81} + \dfrac{8\times 81}{81} = y \)

\( \dfrac{x^2 - 10x + 25}{81} + \dfrac{648}{81} = y \)


\( \dfrac{x^2 - 10x + 25 + 648 }{81} = y \)

Answer 45

so what is 25 + 648 =

Answer 46

25/648

Answer 47

0.03

Answer 48

no

add the numbers

Answer 49

673

Answer 50

\( \dfrac{x^2 - 10x + 673 }{81} = y \)


Now the last step is to change y to \( f^{-1}(x)\)

So your final answer is

\( f^{-1}(x) = \dfrac{x^2 - 10x + 673 }{81}\)

Answer 51

thanks

Answer 52

you're welcome