Question

If we have the functions as $f\left( x \right)={{x}^{2}}-3$ and $g\left( x \right)=5x$, how do you find $f\left( g\left( -3 \right) \right)$?

Answer 1

This question is from the chapter of functions which belongs to the topic pre-calculus. In this question, we will find the value of $f\left( g\left( -3 \right) \right)$. In solving this question, we will first find the value of $f\left( g\left( x \right) \right)$. After that, we will put the value of x as -3. After solving the further question, we will get our required answer.

Complete step-by-step solution:
Let us solve this question.
In this question, we have to find the value of $f\left( g\left( -3 \right) \right)$ using the given data. The given data are:
$f\left( x \right)={{x}^{2}}-3$ and
$g\left( x \right)=5x$
As we can see that $f\left( x \right)={{x}^{2}}-3$, so we will put the value of x as $g\left( x \right)$ in the function of $f\left( x \right)$.
So, after putting the value of x as $g\left( x \right)$ in the equation $f\left( x \right)={{x}^{2}}-3$, we can write
$f\left( g\left( x \right) \right)={{\left( g\left( x \right) \right)}^{2}}-3$
As it is given in the question that the value of $g\left( x \right)$ is $5x$, so we can write the above equation as
$\Rightarrow f\left( g\left( x \right) \right)={{\left( 5x \right)}^{2}}-3$
The above equation can also be written as
$\Rightarrow f\left( g\left( x \right) \right)={{5}^{2}}\times {{x}^{2}}-3$
As we know that the square of 5 is 25, so we can write the equation as
$\Rightarrow f\left( g\left( x \right) \right)=25{{x}^{2}}-3$
Now, it is asked in the question that we have to find the value of $f\left( g\left( -3 \right) \right)$, so we will put the value of x in the above equation as -3.
After putting the value of x as -3 in the above equation, we can write
$\Rightarrow f\left( g\left( -3 \right) \right)=25{{\left( -3 \right)}^{2}}-3$
The above equation can also be written as
$\Rightarrow f\left( g\left( -3 \right) \right)=25\times 9-3$
The above equation can also be written as
$\Rightarrow f\left( g\left( -3 \right) \right)=225-3=222$
So, we get that the value of $f\left( g\left( -3 \right) \right)$ is 222.

Note: We should have a better knowledge in the topic of function to solve this type of question easily.
Whenever we have to find the value of $f\left( g\left( x \right) \right)$, then we will first put the value of x in the equation as $g\left( x \right)$ and solve the further equation, then we will get our answer. Remember that $f\left( g\left( x \right) \right)$ can also be written as $fog\left( x \right)$.