Define the real valued function f:R−{0}→R defined by f(x)=x1, x∈R−{0}. Complete the Table given below using this definition. What is the domain and range of this function? $\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline
x & -2 & -1.5 & -1 & -0.5 & 0.25 & 0.5 & 1 & 1.5 & 2
\hline
y=\frac{1}{x} & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots
\hline
\end{tabular}
\end{center}
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Step-by-Step Solution
Step 1: Understand the Function Definition
The given function is
f(x)=x1
. This means that for any input value
x
, the output f(x) is its reciprocal. The domain is specified as
R−{0}
, which means all real numbers except
0
, because division by zero is undefined.
Step 2: Calculate Function Values for Negative x
We substitute each negative value of
x
from the table into the function
f(x)=x1
to find the corresponding
y
values. For example, when
x=−2
,
f(−2)=−21=−0.5
.
Step 3: Calculate Function Values for Positive x
Similarly, we substitute each positive value of
x
from the table into the function
f(x)=x1
to find the corresponding
y
values. For example, when
x=0.25
,
f(0.25)=01.25=4
.
Step 4: Complete the Table
By combining all the calculated values, we can complete the given table. The values are rounded to two decimal places where necessary.
Step 5: Determine Domain and Range
The domain of the function
f(x)=x1
is given as all real numbers except
0
, because division by zero is undefined. The range of the function is also all real numbers except
0
, because f(x) can never be
0
(since
1/x=0
has no solution) and can take any other real value.