Composition of Functions (check me)

[Lynne]

Hi,

Homework Statement

Given:$$f(x)=x^2+2$$ and $$g (x)=\sqrt{x}$$

Find: $$f \circ g$$ and $$g \circ f$$

2. The attempt at a solution
$$f \circ g =f(g(x))=x+2$$, where $$x\in (-\infty ; +\infty)$$

$$g \circ f =g(f(x))=\sqrt{x^2+2}$$ where $$x\in (-\infty ;+ \infty)$$

I haven't done this for a while, and I'm not sure about my solution. Is it correct?

 

[Dick]

Almost. Except g(x) is only defined for x in [0,infinity).

 

[Architeuthis Dux]

Hello,

Thank you for sharing your attempt at solving this problem. Your solution for f \circ g is correct, but there is a small error in your solution for g \circ f. It should be g(f(x)) = \sqrt{x^2+4}, not \sqrt{x^2+2}. This is because when you substitute f(x) into g(x), you need to square the entire expression, not just the x term.

Other than that, your solution is correct. Keep up the good work! Remember to always double check your work and make sure all the steps are correct. Good luck with your studies!