Question

A function is of the form $f\left( x \right) = ax + b$ . If $\left( {fof} \right)\left( x \right) = 4x + 9$ , find the values of a and b.

Answer 1

Let $f:A \to B$ and $g:B \to C$ be two functions. Then the composition of $f$ and $g$ , denoted by $gof$, is defined as the function $gof:A \to C$ given by $gof\left( x \right) = g\left( {f\left( x \right)} \right),\forall x \in A$.

Clearly, $gof\left( x \right)$ is a function in which variables of the function $g\left( x \right)$ are replaced by $f\left( x \right)$ .To find $\left( {fof} \right)\left( x \right)$ replace x by $ax+b$ in the given function $f\left( x \right) = ax + b$ , we get an equation and equate it to the given value $4x + 9$ and compare the coefficients to get the value a and b.

Complete step-by-step answer:
Given $f\left( x \right) = ax + b$
We have to find $\left( {fof} \right)\left( x \right)$.
Replace x by $ax+b$ and equate it to the value $4x + 9$
$\left( {fof} \right)\left( x \right) = f\left( {f\left( x \right)} \right)$
$f\left( {f\left( x \right)} \right) = f\left( {ax + b} \right)$

$\Rightarrow a\left( {ax + b} \right) + b$

$\Rightarrow {a^2}x + ab + b$

Find values of a and b
It is given that $\left( {fof} \right)\left( x \right) = 4x + 9$
Hence, ${a^2}x + ab + b = 4x + 9$
We can compare the coefficients of $x$ and the constant term on both sides because we are given that $\left( {fof} \right)\left( x \right)$ exactly equals to the $4x + 9$ for all values of $x$ , i.e. $\left( {fof} \right)\left( x \right) = 4x + 9$ is an identity.
Thus, we can compare the coefficients of various powers of $x$.

${a^2} = 4$

$\Rightarrow a = \sqrt 4$

$\because a = \pm 2$

As 2 $\times$ 2 = 4 and also (-2) $\times$ (-2) = 4.

$ab + b = 9$

$\Rightarrow b\left( {a + 1} \right) = 9$

Put values of a to calculate values of b
For a = 2

$b\left( {a + 1} \right) = 9$

$\Rightarrow b\left( {2 + 1} \right) = 9$

$\Rightarrow b \times 3 = 9$

$\Rightarrow b = \dfrac{9}{3}$

$\because b = 3$

For a = -2

$b\left( {a + 1} \right) = 9$

$\Rightarrow b\left( { - 2 + 1} \right) = 9$

$\Rightarrow b \times \left( { - 1} \right) = 9$

$\because b = - 9$

The values of (a, b) are (2, 3) and (-2, -9).

Note: Similarly, $fog\left( x \right)$ is a function in which variables of the function $f\left( x \right)$ are replaced by $g\left( x \right)$.
The composition of functions is also useful in checking the invertible function, hence finding the inverse of the function.
A function $f:X \to Y$ is defined to be invertible, if there exists a function $g:Y \to X$ such that $gof = {I_x}$ and $fog = {I_y}$. The function $g$ is called the inverse of $f$ and is denoted by ${f^{ - 1}}$ .
We should always be cleared between an equation and an identity. An equation has some solutions, it is not satisfied by all the values of the domain of the variable while identity is satisfied by all the values in the domain of the variable.