Question

How do you find the domain and range of $ f(x) = \dfrac{1}{{1 + {x^2}}} $ ?

Answer 1

Hint: To determine the domain of $ f(x) $ ,find out the set of values that the variable x in the function can have and to determine the range express the function as x in terms of y and consider the fact that the range of $ f(x) $ will be the domain of $ x $ .

Complete step-by-step answer:
Given a function $ f(x) = \dfrac{1}{{1 + {x^2}}} $
Domain is basically the set to values that the x in the function can have.
The only restriction that we have is that the denominator should not be equal to zero
Since, $ 1 + {x^2} $ is always greater than zero for all values of x.
\[\,\forall x \in R,1 + {x^2}\, > 0\]
Therefore, the domain of the function $ f(x) $ is $ x \in R $ .
To determine the range of any function ,express the function as $ x $ in terms of $ y $ and the set of values that y can have will be the range of the function .In other words the range of $ f(x) $ will be the domain of $ x $ .
So in our function, expressing it as $ x $ in terms of y
 \[
  y = \dfrac{1}{{1 + {x^2}}} \\
  y(1 + {x^2}) = 1 \\
  y + y{x^2} = 1 \\
  y{x^2} = 1 - y \\
  {x^2} = \dfrac{{1 - y}}{y} \\
  x = \sqrt {\dfrac{{1 - y}}{y}} \;
 \]
Since the range of $ f(x) $ is the domain of $ x $ .
We can say that
 $
  \left( {\dfrac{{1 - y}}{y}} \right) > 0 \\
  y \in {R^*}_ + \\
  1 - y \geqslant 0 \\
  y \leqslant 1 \\
  $
 $ y $ can have values between 0 and 1 excluding 0 i.e. in the interval $ (0.1] $
The range is $ y \in (0.1] $
Therefore, the domain of $ f(x) $ is $ x \in R $ and range of $ f(x) $ is $ y \in (0.1] $
So, the correct answer is “The domain of $ f(x) $ is $ x \in R $ and the range of $ f(x) $ is $ y \in (0.1] $ ”.

Note: DOMAIN: Let R be a relation from a set A to a set B. Then the set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R.
Thus, domain of $ R = \left\{ {a:(a,b) \in R} \right\} $
Clearly, the domain of $ R \subseteq A $ .
If $ A = \left\{ {1,3,5,7} \right\},B = \left\{ {2,4,6,8,10} \right\} $ and $ R = \left\{ {\left( {1,8} \right),\left( {3,6} \right),\left( {5,2} \right),\left( {1,4} \right)} \right\} $ is a relation from A to B,
Then,
Domain $ (R) = \left\{ {1,3,5} \right\} $
RANGE: Let R be a relation from a set A to a set B . Then the set of all second components or coordinates of the ordered pairs belonging to R is called the range of R.
Thus, Range of $ R = \left\{ {b:\left( {a,b} \right) \in R} \right\} $
Clearly, range of $ R \subseteq B $
If $ A = \left\{ {1,3,5,7} \right\},B = \left\{ {2,4,6,8,10} \right\} $ and $ R = \left\{ {\left( {1,8} \right),\left( {3,6} \right),\left( {5,2} \right),\left( {1,4} \right)} \right\} $ is a relation from A to B,
Then,
Range $ (R) = \left\{ {8,6,2,4} \right\} $