Question 202895
(f+g)(x) is simply shorthand for f(x) + g(x). So (f+g)(x) in your problem is 1/x + 1/x^3. Since this is not one of the listed answers we will have to add the fractions to see what it looks like then.<br>
To add fractions:<ol><li>Factor each denominator</li><li>Multiply the numerator and denominator of each fraction by whatever is needed to make each denominator the same</li><li>Simplify (multiply) the numerators <i>but leave the denominators factored</i></li><li>Add the numerators of the fractions</li><li>Factor the new numerator, if possible</li><Cancel common factors (reduce the fraction), if any</li><li>If the denominator is still factored, multiply it out.</ol>
We'll use this on:
{{{1/x + 1/(x^3)}}}
Factor the denominators:
{{{1/x + 1/(x*x*x)}}}
Multiply the top and bottom of each fraction, as needed, to get the denominators the same. It takes some practice to develop an "eye" for how to do this. I hope that it is clear that if the top and bottom of the first fraction is multiplied by (x*x) then its denominator would be the same as the denominator in the second fraction:
{{{((x*x)/(x*x))*(1/x) + 1/(x*x*x)}}}
Multiply the numerator(s):
{{{(x^2)/(x*x*x) + 1/(x*x*x)}}}
Add the fractions:
{{{(x^2 + 1)/(x*x*x)}}}
Factor the numerator. This numerator will not factor.
Cancel common factors. There are no common factors
Multiply out the denominator.
{{{(x^2+1)/(x^3)}}}
This is (f+g)(x). Since it does not match any of the answers you've provided, I can only guess that either you mistyped the answers (the first two <i>are</i> exactly the same!?) or there is an error in the answers provided in your source of the problem.