Question

Let f(x) = x2 - 16. Find f-^1(x).
Please, someone help. i don't need the answer, just help.

Answer 1

does it state that x > 0 or x < 0 or something like that?

Answer 2

no, everything that I see for the problem is what I have above. :/

Answer 3

ok usually they'll add the restriction that x > 0, so let's go with that

Answer 4

f(x) = x^2 - 16

y = x^2 - 16 ... replace f(x) with y

x = y^2 - 16 ... swap x and y

now solve for y and tell me what you get

Answer 5

I'd hate to sound stupid, but would it be along the lines of something like 16x = y^2?

Answer 6

you don't sound stupid

Answer 7

first you need to move that 16 over, so you add 16 to both sides

x = y^2 - 16

x + 16 = y^2

Answer 8

oohhhh okay. Instead I just made it 16x.

Answer 9

my bad. :P

Answer 10

so

y^2 = x + 16

then you take the square root of both sides

because x > 0 is the domain of f(x), this means that the range of f^-1(x) is y > 0 (remember domain and range swap when you go from a function to its inverse)

so because y > 0, this means that we go with the positive square root, so

y = sqrt(x+16)

Answer 11

therefore, if x > 0 is applied to the original function, then the inverse is

\[\large f^{-1}(x) = \sqrt{x+16}\]

Answer 12

That's one of the answers! Thanks so much! But for future reference, what would I have to remember in order to solve equations like these? I just started working with this kind of stuff, and I don't exactly understand it too well.

Answer 13

when solving equations you are following PEMDAS in reverse

so normally if you started with something like 2*3 + 5, you follow PEMDAS in the forward direction to get

2*3+5

6+5

11

so 2*3+5 = 11

now say that the '3' is covered up and turned into an 'x' to get 2x+5 = 11

-----------------

now you follow PEMDAS in reverse to undo all the operations to get x all by itself

2x + 5 = 11

2x = 11 - 5 ... (undoing addition)

2x = 6

x = 6/2 .... (undoing multiplication)

x = 3

Answer 14

notice how multiplication normally comes first in PEMDAS

but because we're following it in reverse order, addition comes first (ie that's the first thing to undo)

Answer 15

so looking at x = y^2 - 16

the first thing to undo is the subtraction of 16, so we add 16 to both sides to get

x+16 = y^2

y^2 = x+16

then we undo the exponent by taking the square root of both sides

y = sqrt(x+16)

Answer 16

normally exponentiation is done before subtraction (since the E comes before S), but again we're going in reverse when solving for a variable

Answer 17

ohhh, so I have to find the square root, only if there are exponents. I was so confused on when it was necessary to find the square root. thanks, that clears things up a little more for me. (:

Answer 18

only if the exponent is 2

ex:

x^2 = 4

x = sqrt(4) or x = -sqrt(4)

x = 2 or x = -2


but if the exponent is 3 like in x^3 = 8, then you use a cube root to solve for x

Answer 19

yeah, that makes sense. I just started so I've only seen exponents of 2 so far, but that's good to know for future reference.(:

Answer 20

ok glad it's making more sense

Answer 21

If you're not busy, can I have your help with one more question?

Answer 22

sure go for it

Answer 23

Everything you've explained to me, makes complete sense. But...
Let f(x) = x - 2 and g(x) = x2 - 7x - 9. Find f(g(-1)).

Answer 24

first we need to find g(-1)

Answer 25

what is g(-1)

Answer 26

I have no idea, I'm sorry. :( would i just solve the equation g(x) = x2 - 7x - 9?

Answer 27

g(-1) means plug x = -1 into g(x)

Answer 28

g(x) = x^2 - 7x - 9

g(-1) = (-1)^2 - 7(-1) - 9

g(-1) = ??

Answer 29

g(-1) = 1+7-9
am i on the right track?

Answer 30

yep, keep going

Answer 31

Then g(-1)+9=8, right?

Answer 32

why not just combine 1+7-9 to get ______

Answer 33

ohhh, hahaha that's obvious, sorry. -1 :)

Answer 34

so g(-1) = -1, good

Answer 35

so f(g(-1)) turns into f(-1)

since you replace g(-1) with -1 (we've just shown that g(-1) = -1)

Answer 36

now you must find f(-1)

Answer 37

f(x) = x - 2
f(-1) = (-1) - 2
f(-1) = -3

Answer 38

very good, that means

f(g(-1)) = -3

Answer 39

oh wow!! that's it?

Answer 40

yep, that's all there is to it

Answer 41

hahaha i guess it just looked over whelming. thank you so much again. :) i really really appreciate it.

Answer 42

you're welcome, I'm glad I could help