Question

Find the range of the function \[f(x) = 2 - 3x\], \[x \in R\], \[x > 0\].

Answer 1

Hint: The range of a function \[f(x)\] refers to the possible values that a function can attain for the values of \[x\] which are in the domain of \[f\left( x \right)\].

Complete step by step solution:
The domain of a function \[f(x)\] refers to the set of values of \[x\] for which \[f\left( x \right)\] is defined. It is given that \[f\left( x \right)\] is defined for all \[x > 0\]. Hence the domain of \[f\left( x \right)\] is \[\left( {0,\infty } \right)\].
To find the range of a function we need to find the values of \[f(x)\] for all \[x\] belonging to the domain of \[x\].
Given:
\[x > 0\]
We need to transform the \[x\] on the left hand side to \[2 - 3x\], for that multiply both sides by \[3\]:
\[ \Rightarrow 3x > 0\]
Multiply both sides by \[ - \] sign:
\[ \Rightarrow - 3x < 0\]
Note that the sign changed from > to < as we multiplied by \[ - \] on both sides.
Add \[2\] to both sides:
\[ \Rightarrow 2 - 3x < 2\]

Therefore the range of \[f\left( x \right)\] is \[( - \infty ,2)\].

Note:
For solving any problems like this where the domain and range needs to be found, first the definition of the given function needs to be understood. First the domain of the function must be found following the definition then the range can be easily found accordingly. For example if a function is like \[f(x) = \sqrt {4 - x} \], then by the definition of roots any number under root will always be positive. So for the domain \[4 - x\] must always be greater than \[0\] . In this way the definitions of various functions need to be analysed.