The answer is:
[tex]f(x)+g(x)=3x^{2} +\frac{3x}{2}-9[/tex]
We are working with function addition, to add or subtract two o more functions, we need to follow the following form:
[tex](f+g)=f(x)+g(x)[/tex]
To simplify the expression, we need to work with the like terms, like terms are the terms that share the same variable and the same coefficient, for example:
[tex]x+3x+x^{2}=x^{2} +4x[/tex]
We were able to add only the first two terms since the third term does not share the exponent with the other two.
We are given the functions:
[tex]f(x)=\frac{x}{2}-3\\\\g(x)=3x^{2}+x-6[/tex]
So, solving, we have:
[tex]f(x)+g(x)=(\frac{x}{2} -3)+(3x^{2} +x-6)\\\\f(x)+g(x)=3x^{2}+\frac{x}{2}+x-3-6\\\\f(x)+g(x)=3x^{2}+\frac{x+2x}{2}-9\\\\f(x)+g(x)=3x^{2} +\frac{3x}{2}-9[/tex]
Hence, the answer is:
[tex]f(x)+g(x)=3x^{2} +\frac{3x}{2}-9[/tex]
Have a nice day!
