Question

The function $f:C\to C$ defined by $f(x)=\dfrac{ax+b}{cx+d}$ for $x\in C$ where $bd\ne 0$ reduces to constant function if
A.$a=c$
B.$b=d$
C.$ad=bc$
D.$ab=cd$

Answer 1

Hint: We are given, the function $f:C\to C$ defined by $f(x)=\dfrac{ax+b}{cx+d}$ for $x\in C$ where $bd\ne 0$. Assume the constant to be $m$ and simplify it. Functions are relations where each input has a particular output.

Complete step-by-step answer:
We are given $f(x)=\dfrac{ax+b}{cx+d}$.
Now let us assume the constant function to be $m$.
We get,
$f(x)=\dfrac{ax+b}{cx+d}=m$
Now cross multiplying we get,
$\Rightarrow$ $ax+b=m(cx+d)$
Simplifying we get,
$\Rightarrow$ $ax+b=mcx+md$
Now comparing the equation we get,
$\Rightarrow$ $a=mc$ and $b=md$
Now, for $a=mc$ , we can write it as,
$\Rightarrow$ $\dfrac{a}{c}=m$ …………. (1)
Also, for $b=md$, we can write it as,
$\Rightarrow$ $\dfrac{b}{d}=m$ ………… (2)
From (1) and (2), we get,
$\Rightarrow$ $\dfrac{a}{c}=\dfrac{b}{d}=m$
Hence, we get,
$\Rightarrow$ $ad=bc$
Therefore, the function $f:C\to C$ defined by $f(x)=\dfrac{ax+b}{cx+d}$ for $x\in C$ where $bd\ne 0$ reduces to constant function if $ad=bc$.
The correct answer is option (C).

Additional information:
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and codomain or range. A function is generally denoted by $f(x)$ where $x$ is the input. The general representation of a function is $y=f(x)$. A function in maths is a special relationship among the inputs (i.e. the domain) and their outputs (known as the codomain) where each input has exactly one output and the output can be traced back to its input. A function is an equation for which any $x$ that can be put into the equation will produce exactly one output such as $y$ out of the equation. It is represented as $y=f(x)$.

Note: In the problem we have assumed the constant function to be $m$. A function is a relation which describes that there should be only one output for each input (or) we can say that a special kind of relation (a set of ordered pairs) which follows a rule i.e. every $x$-value should be associated with only one $y$-value is called a function.