Find an expression of g(x) in terms of x for an equation f(x)?

[fs93]

Homework Statement

The function f is defined by f(x) = x^3

Find an expression for g(x) in terms of x in each of the following cases.

(a) f [ g(x) ] = x+1

(b) g [ f(x) ] = x+1

Homework Equations

The Attempt at a Solution

I got the same answer both times. g(x) = cubic route (x+1). Because if I put the cubic route of x+1 in f(x), it is cubed, and I'm left with x+1.

For (b) its the same. If I have cubic route of (x+1), and then cube it, I'm left with x+1. Is this correct?

I'd appreciate some help :)

 

[JG89]

The first one is okay. For the second one, if $$g(x) = \sqrt[3]{x + 1}$$ then $$g(f(x)) = \sqrt[3]{x^3 + 1} \neq x + 1$$.

Why don't you try $$g(x) = \sqrt[3]{x} + 1$$

 

[fs93]

Thanks, that really clears things up.

But is there a method that I can always apply to such a question?

Or do you always have to keep doing trial and logic until you find the correct function?

 

[JG89]

For the type of question you posted, it's more or less using your logic and just seeing what the correct function should be, combined with some trial and error.

 

[fs93]

For the type of question you posted, it's more or less using your logic and just seeing what the correct function should be, combined with some trial and error.

That's what I'm going to practice then. Thanks a million.

 

[Mark44]

For b, you want to find g so that g(x³ + 1) = x + 1, so you need to figure out what g needs to do to an input value so that the output is x + 1.

To get from x³ + 1 to x + 1, g would need to:

1. Subtract 1 from the input value.
2. Take the cube root (not route) of the value from step 1.
3. Add 1 to the value from step 2.

JG89's suggestion, $$g(x) = \sqrt[3]{x} + 1$$
doesn't do the first step, just the second and third, so doesn't work as the formula for g(x).