Question

If g(x) = 3x + 5, what is g-1(x)?

Answer 1

1. Let y = g(x). After doing so, the equation becomes
y = 3x + 5

2. Swap x and y
x = 3y + 5

3. Isolate y:
\[\frac{x - 5}{3} = y\]

4. Rewrite as

\[y = \frac{x - 5}{3}\]

5. Replace y with \(g^{-1}(x)\):

\[g^{-1}(x) = \frac{x - 5}{3}\]

Answer 2

Oh that's what the -1 means, that's right. Thanks, Hero.

Answer 3

-1 in the exponent always means "inverse"

Answer 4

You pronounce \(g^{-1}(x)\) as "g inverse of x"...as I'm sure you already know.

Answer 5

Can you show me the steps between step 2 and step 3?

Answer 6

Nevermind, I get it. XD

Answer 7

Okay. I try to make my steps clear.

Answer 8

I know some steps may be unclear. It is good that you try to understand then ask questions if you don't. Most students would just stare at it without speaking. Then leave assuming that I will only confuse them more.

Answer 9

I assure you that I have the ability to make things that are confusing clearer and not more confusing.

Answer 10

Oh no doubt. It just took me a moment to piece together how to solve for y. That's all. :)

Answer 11

listen to him^^^:)

Answer 12

Thanks for the support @zzr0ck3r. You got back pretty quick bro.

Answer 13

One more question,

What is the description of f(x) = -3 - x^6?


The exponent is 6 so it's an even graph, and the 3 is a negative so it curves to the left?

Answer 14

Err *decreases to the left

Answer 15

When figuring this out, you want to begin with the lead term, which is x^6. So what you do is say that f(x) is the graph of x^6 flipped over the horizontal axis and decreased by 3.

Answer 16

If you graph f(x), you will see that this is true.

Answer 17

But it is an even function?

Answer 18

You can also mention that it is an even function. You figure that out by replacing x with -x and then simplifying:

f(-x) = -((-x)^6 - 3
f(-x) = - x^6 -3

You know it's an even function when f(-x) = f(x)

Answer 19

So you don't focus on the exponent when determining this?

Answer 20

When determining it, you focus on the definition of an even function. That is f(x) = f(-x). That's how you determine it. Focusing on the exponent is like making assumptions. Many people make assumptions with mathematics and that's what leads to mistakes. What if you had:

f(x) = x^2 - x

Now let's replace that with f(-x) and see what we get:
f(-x) = (-x)^2 -(-x)
f(-x) = x^2 + x

We did not arrive back at the original expression even though we had an even exponent.

Answer 21

And by not arriving back at the original equation, it's an odd function?

Answer 22

It's only an odd function if f(-x) = -f(x)
-f(x) = -(x^2 - x) = -x^2 + x

So it's not an odd function either.

Answer 23

It is neither even nor odd. So as you can see, we have to use the definition of even and odd function to determine what kind of function it is. Making assumptions will not work. Mathematics is solved by using definitions, properties, postulates, and theorems. You're allowed to use assumptions when proving theorems and such, but with computation, you have to be careful.

Answer 24

When you make assumptions in mathematics, they have to be provable. Otherwise they are of no good use.

Answer 25

Got it, I understand now. I don't even remember learning how to tell whether it was even or odd. But I'm glad this got cleared up before my final. Thanks for the help!

Answer 26

yw. @zzr0ck3r did you want to add anything?

Answer 27

Okay, well, I'm out. ttyl

Answer 28

Goodnight. Thank you again :)

Answer 29

Bye

Answer 30

@hero not at all, I was just watching and eating and agree with you:)

Answer 31

I figured you must have been eating.