Given $f(x)=x-1$ and $g(x)=5x^2$, first find $f+g,f-g,fg$, and $\frac{f}{g}$. Then determine the domain for each function.
$(f+g)(x)=5x^2+x-1$ (Simplify your answer.)

Step 1 ：Given $f(x)=x-1$ and $g(x)=5x^2$. We need to find the sum, difference, product, and quotient of the functions $f(x)$ and $g(x)$. The domain of each function is the set of all real numbers for which the function is defined.

Step 2 ：For the sum and difference of the functions, the domain is the intersection of the domains of $f(x)$ and $g(x)$. Since both $f(x)$ and $g(x)$ are polynomials, their domains are all real numbers. Therefore, the domain of $(f+g)(x)$ and $(f-g)(x)$ is all real numbers.

Step 3 ：For the product of the functions, the domain is also the intersection of the domains of $f(x)$ and $g(x)$. Again, since both $f(x)$ and $g(x)$ are polynomials, their domains are all real numbers. Therefore, the domain of $(fg)(x)$ is all real numbers.

Step 4 ：For the quotient of the functions, the domain is the set of all real numbers for which the denominator $g(x)$ is not equal to zero. We need to solve the equation $g(x)=0$ to find the values of $x$ that are not in the domain of $\left(\frac{f}{g}\right)(x)$.

Step 5 ：$f+g(x)=5x^2+x-1$ with domain all real numbers.

Step 6 ：$f-g(x)=-5x^2+x+1$ with domain all real numbers.

Step 7 ：$fg(x)=5x^3-5x^2$ with domain all real numbers.

Step 8 ：$\left(\frac{f}{g}\right)(x)=\frac{x-1}{5x^2}$ with domain all real numbers except $x=0$.

Step 9 ：$\boxed{{\displaystyle \text{The sum of the functions }f(x)\text{ and }g(x)\text{ is }(f+g)(x)=5x^2+x-1\text{ with domain all real numbers.}}}$

Step 10 ：$\boxed{{\displaystyle \text{The difference of the functions }f(x)\text{ and }g(x)\text{ is }(f-g)(x)=-5x^2+x+1\text{ with domain all real numbers.}}}$

Step 11 ：$\boxed{{\displaystyle \text{The product of the functions }f(x)\text{ and }g(x)\text{ is }(fg)(x)=5x^3-5x^2\text{ with domain all real numbers.}}}$

Step 12 ：$\boxed{{\displaystyle \text{The quotient of the functions }f(x)\text{ and }g(x)\text{ is }\left(\frac{f}{g}\right)(x)=\frac{x-1}{5x^2}\text{ with domain all real numbers except }x=0.}}$