Question

Combining Functions Find $f+g,f-g,fg$and $\frac{f}{g}$and their domains.

$f\left(x\right)=x^2+x,g\left(x\right)=x^2$

Short answer

The required functions and their domains are:

The domain of$\left(f+g\right)\left(x\right)=2x^2+x$ is the set of all real numbers.

The domain of$\left(f-g\right)\left(x\right)=x$ is the set of all real numbers.

The domain of$\left(fg\right)\left(x\right)=x^4+x^3$ is the set of all real numbers.

The domain of$\left(\frac{f}{g}\right)\left(x\right)=1+\frac{1}{x}$ is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$.

Step-by-step solution

– Domain of combined functions.

Domain of a function is the set of all possible input values.

If domains of f and g are A and B then,

• Domain of$f+g$ is $A\cap B$.
• Domain of $f-g$is $A\cap B$.
• Domain of $fg$is $A\cap B$.
• Domain of$\frac{f}{g}$ is $\left\{x\in A\cap B|g\left(x\right)\ne 0\right\}$.

– Given information.

The given functions are:

$\begin{array}{c}f\left(x\right)=x^2+x\\ g\left(x\right)=x^2\end{array}$

Both functions are polynomials. So, the domains of the given functions are the set of all real numbers.

– Sum of given functions and its domain.

$\begin{array}{c}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ =\left(x^2+x\right)+x^2\\ =2x^2+x\end{array}$

The intersection of domains of both functions f and g is the set of all real numbers. So, the domain of$\left(f+g\right)$ is the set of all real numbers.

– Difference of given functions and its domain.

$\begin{array}{c}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ =\left(x^2+x\right)-x^2\\ =x\end{array}$

The intersection of domains of both functions f and g is the set of all real numbers. So, the domain of$\left(f-g\right)$ is the set of all real numbers.

– Product of given functions and its domain.

$\begin{array}{c}\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)\\ =\left(x^2+x\right)\left(x^2\right)\\ =x^4+x^3\end{array}$

The intersection of domains of both functions f and g is the set of all real numbers. So, the domain of$\left(fg\right)$ is the set of all real numbers.

– Quotient of given functions and its domain.

$\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{x^2+x}{x^2}\\ =\frac{x^2}{x^2}+\frac{x}{x^2}\\ =1+\frac{1}{x}\end{array}$

The intersection of domains of both functions f and g is the set of all real numbers and the function $g\left(x\right)=0$if $x^2=0$, that is $x=0$. So, the domain of $\left(\frac{f}{g}\right)$is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$.

Thus, the required functions are their domains are:

The domain of$\left(f+g\right)\left(x\right)=2x^2+x$ is the set of all real numbers.

The domain of$\left(f-g\right)\left(x\right)=x$ is the set of all real numbers.

The domain of$\left(fg\right)\left(x\right)=x^4+x^3$ is the set of all real numbers.

The domain of$\left(\frac{f}{g}\right)\left(x\right)=1+\frac{1}{x}$ is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$.