Question

If \[g\left( x \right) = {x^2} + x - 1\;\] and \[\left( {gof} \right)\left( x \right) = 4{x^2} - 10x + 5\], then \[f\left( {\dfrac{5}{4}} \right)\;\] is equal to:
A) $\dfrac{3}{2}$
B) $\dfrac{1}{2}$
C) $ - \dfrac{3}{2}$
D) $ - \dfrac{1}{2}$

Answer 1

Hint: A composite function is defined as a function that depends on another function. A composite function has been created in such a way that one function is substituted into another function.
For example, \[f\left( {g\left( x \right)} \right)\] is the composite function that is formed when \[g\left( x \right)\] is substituted for \[x\] in \[f\left( x \right)\].
\[f\left( {g\left( x \right)} \right)\] is read as “f of g of x”.
\[f\left( {g\left( x \right)} \right)\] can also be written as \[(f \circ g)\left( x \right)\] or \[fg\left( x \right)\],
In the composition \[(f \circ g)\left( x \right)\], the domain of \[f\] becomes \[g\left( x \right)\].
To solve a composite function, we need to find the composition of two functions. We don’t use a multiplying sign or a dot but instead of that we use a small circle \[( \circ )\] for the composition of a function. Here is the list of steps to solve a composite function:
Rewrite the composition in a different form.
For example
\[(f \circ g){\text{ }}\left( x \right){\text{ }} = {\text{ }}f{\text{ }}\left[ {g{\text{ }}\left( x \right)} \right]\]
Substitute the variable \[x\] that is in the outside function with the inside function.
Simplify the function.

Complete step-by-step answer:
We have been given in the question \[g\left( x \right) = {x^2} + x - 1\;\] and \[\left( {gof} \right)\left( x \right) = 4{x^2} - 10x + 5\].
We have to find the value of \[f\left( {\dfrac{5}{4}} \right)\;\].
Now first we will try to simplify the equation \[\left( {gof} \right)\left( x \right) = 4{x^2} - 10x + 5\]
$ \Rightarrow \left( {gof} \right)\left( x \right) = 4{x^2} - 10x + 5$
We will try to make the above equation similar to \[g\left( x \right) = {x^2} + x - 1\;\]
$ \Rightarrow \left( {gof} \right)\left( x \right) = {\left( {2x} \right)^2} - 8x + 4 - 2x + 1$
Now we will use the property ${(a - b)^2} = {a^2} - 2ab + {b^2}$
$ \Rightarrow \left( {gof} \right)\left( x \right) = {\left( {2x - 2} \right)^2} - 2x + 2 - 1$
$ \Rightarrow \left( {gof} \right)\left( x \right) = {\left( {2x - 2} \right)^2} - \left( {2x - 2} \right) - 1$
We will make the equation as close as possible to \[g\left( x \right) = {x^2} + x - 1\;\].
$ \Rightarrow \left( {gof} \right)\left( x \right) = {\left( {2 - 2x} \right)^2} + \left( {2 - 2x} \right) - 1$
When we compare the above equation with \[g\left( x \right) = {x^2} + x - 1\;\], we get
$ \Rightarrow f(x) = 2 - 2x$
Now substituting the value of \[f\left( {\dfrac{5}{4}} \right)\;\], we get
$ \Rightarrow f\left( {\dfrac{5}{4}} \right) = 2 - 2\left( {\dfrac{5}{4}} \right)$
Taking the LCM of RHS and simplifying further, we get
$ \Rightarrow f\left( {\dfrac{5}{4}} \right) = \left( {\dfrac{{8 - 10}}{4}} \right) = \dfrac{{ - 2}}{4} = - \dfrac{1}{2}$
If \[g\left( x \right) = {x^2} + x - 1\;\] and \[\left( {gof} \right)\left( x \right) = 4{x^2} - 10x + 5\], then \[f\left( {\dfrac{5}{4}} \right)\;\] is equal to $ - \dfrac{1}{2}$.

So, option (D) is the correct answer.

Note: The order in the composite function is the most important because \[(f \circ g){\text{ }}\left( x \right)\] is NOT equal to \[(g \circ f){\text{ }}\left( x \right)\] and cannot be used as vice versa.
"Composite function" is defined as the method in which we apply one function to the results of another.
First apply \[f()\], then apply \[g()\]
The domain of the first function has respect when calculating the domain.
Some functions can be factorized or factored and expanded into two (or more) simpler functions.