Question

How do you find $f\left( x \right)+g\left( x \right)$ , $f\left( x \right)-g\left( x \right)$, $f\left( x \right).g\left( x \right)$ , $\left( \dfrac{f}{g} \right)\left( x \right)$ given that $f\left( x \right)=\dfrac{3}{x-7}$ and $g\left( x \right)={{x}^{2}}+5x$ ?

Answer 1

Hint: Whenever we come across these types of questions where there are arithmetic operations to be performed on certain functions of a kind, we just need to perform the same arithmetic operation on the given value of the functions.

Complete step by step solution:
The given functions are, $f\left( x \right)=\dfrac{3}{x-7}$ and $g\left( x \right)={{x}^{2}}+5x$
The first part is to find $f\left( x \right)+g\left( x \right)$
The arithmetic operation which is performed between these two functions is addition.
To evaluate we apply the same arithmetic operation on the value of the functions.
It is given by,
$\Rightarrow f\left( x \right)+g\left( x \right)=\dfrac{3}{x-7}+{{x}^{2}}+5x$
Now let us simplify the expression to write it in its lowest terms.
$\Rightarrow f\left( x \right)+g\left( x \right)=3+{{x}^{2}}\left( x-7 \right)+5x\left( x-7 \right)$
$\Rightarrow 3+{{x}^{3}}-7{{x}^{2}}+5{{x}^{2}}-35x$
$\Rightarrow {{x}^{3}}-2{{x}^{2}}-35x+3$
Hence, $f\left( x \right)+g\left( x \right)={{x}^{3}}-2{{x}^{2}}-35x+3$
The second part is to find $f\left( x \right)-g\left( x \right)$
The arithmetic operation which is performed between these two functions is subtraction.
To evaluate we apply the same arithmetic operation on the value of the functions.
It is given by,
$\Rightarrow f\left( x \right)-g\left( x \right)=\dfrac{3}{x-7}-\left( {{x}^{2}}+5x \right)$
Now let us simplify the expression to write it in its lowest terms.
$\Rightarrow f\left( x \right)+g\left( x \right)=3-{{x}^{2}}\left( x-7 \right)-5x\left( x-7 \right)$
$\Rightarrow 3-{{x}^{3}}+7{{x}^{2}}-5{{x}^{2}}+35x$
$\Rightarrow -{{x}^{3}}+2{{x}^{2}}+35x+3$
Hence, $f\left( x \right)-g\left( x \right)=-{{x}^{3}}+2{{x}^{2}}+35x+3$
The third part is to find $f\left( x \right).g\left( x \right)$
The arithmetic operation which is performed between these two functions is multiplication.
To evaluate we apply the same arithmetic operation on the value of the functions.
It is given by,
$\Rightarrow f\left( x \right)\times g\left( x \right)=\dfrac{3}{x-7}\times \left( {{x}^{2}}+5x \right)$
Now let us simplify the expression to write it in its lowest terms.
$\Rightarrow f\left( x \right)\times g\left( x \right)=\dfrac{3\left( {{x}^{2}}+5x \right)}{x-7}$
$\Rightarrow \dfrac{\left( 3{{x}^{2}}+15x \right)}{x-7}$
Hence, $f\left( x \right).g\left( x \right)=\dfrac{\left( 3{{x}^{2}}+15x \right)}{x-7}$
The fourth part is to find $\left( \dfrac{f}{g} \right)\left( x \right)$
The arithmetic operation which is performed between these two functions is division.
To evaluate we apply the same arithmetic operation on the value of the functions.
It is given by,
$\Rightarrow \left( \dfrac{f}{g} \right)x=\dfrac{\left( \dfrac{3}{x-7} \right)}{\left( {{x}^{2}}+5x \right)}$
Now let us simplify the expression to write it in its lowest terms.
$\Rightarrow \left( \dfrac{f}{g} \right)x=\dfrac{3}{\left( {{x}^{2}}+5x \right)\left( x-7 \right)}$
Hence, $\left( \dfrac{f}{g} \right)x=\dfrac{3}{\left( {{x}^{2}}+5x \right)\left( x-7 \right)}$

Note: The function defines a property or a relation between the input and the output such that each input relates to exactly one output. This means that if the object $x$ is in the set of inputs (called the domain) then a function $f$ will map the object $x$ to exactly one object $f\left( x \right)$ in the set of possible outputs (called the codomain).