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

Hint: 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 \[fog\left( x \right)\].