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

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.