Question

If $ f\left( x \right) = {x^3} - 7{x^2} + 5x $ , how do you find all the values for $ f\left( x \right) = - 1 $ ?

Answer 1

Hint: We are required to find the value of a function for a certain value of the variable. This question requires us to have the knowledge of basic and simple algebraic rules and operations such as substitution, addition, multiplication, subtraction and many more like these. A thorough understanding of functions and its applications can be of great significance.

Complete step-by-step answer:
In the given question, we are required to find the value of a variable for a certain value of function. The value of a function at a certain value of variable is found by substituting the value of variable as specified in the question into the function.
So, the function given to us is: $ f\left( x \right) = {x^3} - 7{x^2} + 5x $ .
So, we have to find the value of variable x for which the given function acquires $ \left( { - 1} \right) $ as it’s value.
Hence, $ f\left( x \right) = {x^3} - 7{x^2} + 5x $
So, $ {x^3} - 7{x^2} + 5x = - 1 $ is the condition for finding values of x.
By hit and trial, putting x as $ 1 $ , we get,
 $ f\left( 1 \right) = {1^3} - 7{\left( 1 \right)^2} + 5\left( 1 \right) $
\[ \Rightarrow f\left( 1 \right) = - 1\]
Hence, the value of function is \[\left( { - 1} \right)\] for x equals $ 1 $ .
So, we get one root of the equation $ {x^3} - 7{x^2} + 5x + 1 = 0 $ is $ 1 $ .
Therefore, $ {x^3} - 7{x^2} + 5x + 1 = 0 $
 $ \Rightarrow {x^2}\left( {x - 1} \right) - 6x\left( {x - 1} \right) - \left( {x - 1} \right) = 0 $
 $ \Rightarrow \left( {{x^2} - 6x - 1} \right)\left( {x - 1} \right) = 0 $
So, either $ \left( {{x^2} - 6x - 1} \right) = 0 $ or $ \left( {x - 1} \right) = 0 $ ,
either $ {\left( {x - 3} \right)^2} - 10 = 0 $ or $ \left( {x - 1} \right) = 0 $ ,
either $ x = 3 \pm \sqrt {10} $ or $ x = 1 $ .
Hence, the values of x for which $ f\left( x \right) = - 1 $ are: $ x = 3 + \sqrt {10} $ , $ x = 3 - \sqrt {10} $ and $ x = 1 $ .
So, the correct answer is “ $ x = 3 + \sqrt {10} $ , $ x = 3 - \sqrt {10} $ and $ x = 1 $ ”.

Note: In such questions, we are required to find the values of the variable for which the function acquires the value specified in the question. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable.