Question

If $ f\left( x \right) = {x^2} - 2x + 3 $ , how do you find $ \dfrac{{f\left( {a + h} \right) - f\left( a \right)}}{h} $ ?

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 function for a certain value of variable. 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, we need to replace the variable in the function given to us in the question by the value specified.
So, the function given to us is: $ f\left( x \right) = {x^2} - 2x + 3 $ .
We are required to find the value of the required expression by replacing the value of variable x in the function.
Hence, $ f\left( x \right) = {x^2} - 2x + 3 $
So, $ f\left( a \right) = {a^2} - 2a + 3 $
Also, $ f\left( {a + h} \right) = {\left( {a + h} \right)^2} - 2\left( {a + h} \right) + 3 $
 $ \Rightarrow f\left( {a + h} \right) = {a^2} + 2ah + {h^2} - 2a - 2h + 3 $
Now, $ \dfrac{{f\left( {a + h} \right) - f\left( a \right)}}{h} $
 $ \Rightarrow \dfrac{{\left( {{a^2} + 2ah + {h^2} - 2a - 2h + 3} \right) - \left( {{a^2} - 2a + 3} \right)}}{h} $
 $ \Rightarrow \dfrac{{{a^2} + 2ah + {h^2} - 2a - 2h + 3 - {a^2} + 2a - 3}}{h} $
 $ \Rightarrow \dfrac{{2ah + {h^2} - 2h}}{h} $
 $ \Rightarrow 2a + h - 2 $
Hence, we get the value of the required expression as $ \left( {2a + h - 2} \right) $ by replacing the variable in the original function, that is x, by the specified values and following the algebraic rules.
So, the correct answer is “2a + h - 2”.

Note: Such questions that require just simple change of variable can be solved easily by keeping in mind the algebraic rules such as substitution and transposition. 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.