Question

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

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

Short answer

The required functions and their domains are:

The domain of $\left(f+g\right)\left(x\right)=3x$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 $\left(fg\right)\left(x\right)=2x^2$ is the set of all real numbers.

The domain of$\left(\frac{f}{g}\right)\left(x\right)=\frac{1}{2}$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:

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

Both functions are linear. So, the domain of both functions is the set of all real numbers.

– Sum of given functions and its domain.

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

This function is a linear function. So, domain of this function is the set of all real numbers.

– Difference of given functions and its domain.

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

This function is a linear function. So, domain of this function is the set of all real numbers.

– Product of given functions and its domain.

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

This function is a quadratic function. So, domain of this function is the set of all real numbers.

– Quotient of given functions and its domain.

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

This functions defined for all real values of x except the value for which $2x=0$, that is $x=0$. So, the domain of the above function is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$.

Thus, the required functions are their domains are:

The domain of $\left(f+g\right)\left(x\right)=3x$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 $\left(fg\right)\left(x\right)=2x^2$ is the set of all real numbers.

The domain of$\left(\frac{f}{g}\right)\left(x\right)=\frac{1}{2}$is$\left(-\infty ,0\right)\cup \left(0,\infty \right)$