Hi, I'm having trouble finding the domains on any question like this one:
Find f + g, f − g, fg, and f/g and their domains.
\(\displaystyle f(x)\, =\, \sqrt{9\, -\, x^2\,}\)
\(\displaystyle g(x)\, =\, \sqrt{1\, +\, x\,}\)
Of course the questions would be:
\(\displaystyle (f\, +\, g)(x)\, =\, \sqrt{9\, -\, x^2\,}\, +\, \sqrt{1\, +\, x\,}\)
\(\displaystyle (f\, -\, g)(x)\, =\, \sqrt{9\, -\, x^2\,}\, -\, \sqrt{1\, +\, x\,}\)
\(\displaystyle (fg)(x)\, =\, \sqrt{9\, +\, 9x\, -\, x^3\, -\, x^2\,}\)
\(\displaystyle \left(\frac{f}{g}\right)(x)\, =\, \frac{\sqrt{9\, -\, x^2\,}}{\sqrt{1\, +\, x\,}}\)
I have solved all of them and I understand how to find all of these except for their domains. As far as I know for the (f+g)(x) answer, you would solve for x on both problems.
9-x^2=0 and 1+x=0 and get: x=-(-x-3)(x-3), x=-1 Right?
I think you have to put them into an inequality and that's what I don't know how to do or which way the inequality should face. Or even if it should include an "or equals to" as well. I just don't know. I do understand how to test the inequality and figure out if it's included on a number line but if it's already includes an equals to then that gives you your answer right there. I know how to graph them and all of that but as far as how to put them into an inequality to find their domains, I am lost!
Sorry if this is confusing. Thank you in advance for your help.
Find f + g, f − g, fg, and f/g and their domains.
\(\displaystyle f(x)\, =\, \sqrt{9\, -\, x^2\,}\)
\(\displaystyle g(x)\, =\, \sqrt{1\, +\, x\,}\)
Of course the questions would be:
\(\displaystyle (f\, +\, g)(x)\, =\, \sqrt{9\, -\, x^2\,}\, +\, \sqrt{1\, +\, x\,}\)
\(\displaystyle (f\, -\, g)(x)\, =\, \sqrt{9\, -\, x^2\,}\, -\, \sqrt{1\, +\, x\,}\)
\(\displaystyle (fg)(x)\, =\, \sqrt{9\, +\, 9x\, -\, x^3\, -\, x^2\,}\)
\(\displaystyle \left(\frac{f}{g}\right)(x)\, =\, \frac{\sqrt{9\, -\, x^2\,}}{\sqrt{1\, +\, x\,}}\)
I have solved all of them and I understand how to find all of these except for their domains. As far as I know for the (f+g)(x) answer, you would solve for x on both problems.
9-x^2=0 and 1+x=0 and get: x=-(-x-3)(x-3), x=-1 Right?
I think you have to put them into an inequality and that's what I don't know how to do or which way the inequality should face. Or even if it should include an "or equals to" as well. I just don't know. I do understand how to test the inequality and figure out if it's included on a number line but if it's already includes an equals to then that gives you your answer right there. I know how to graph them and all of that but as far as how to put them into an inequality to find their domains, I am lost!
Sorry if this is confusing. Thank you in advance for your help.
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