Finding the Rule for f in Composition of Functions

[nicnicman]

Homework Statement

Suppose f : R→R and g : R→R where g(x) = 3x + 5 and g o f(x) = x^2 - 7.
Find the rule for f.

Homework Equations

The Attempt at a Solution

These seem pretty simple, but I want to make sure I'm on the right track.

3(f(x)) + 5 = x^2 - 7
3(f(x)) = x^2 - 12
f(x) = (x^2 - 12)/3

How does this look?

 

[Skins]

Looks correct. You could re-write f(x) as

$$f(x) \, = \, \frac{1}{3}x^{2} \, - \, 4$$

but otherwise you are correct and is fine as you have written it.

 

[nicnicman]

Okay, thanks.