Define the function f:R→R by y=f(x)=x2,x∈R. Complete the Table given below by using this definition. What is the domain and range of this function? Draw the graph of f. $\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4
\hline
y=f(x)=x^{2} & & & & & & & & &
\hline
\end{tabular}
\end{center}
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Step-by-Step Solution
Step 1: Understand the Function Definition
The problem defines a function
f
from the set of real numbers
R
to the set of real numbers
R
, given by the equation
y=f(x)=x2
. This means for any input value
x
, the output
y
is the square of
x
.
Step 2: Complete the Table
To complete the table, we substitute each given
x
value into the function
f(x)=x2
and calculate the corresponding
y
value. For example, if
x=−4
, then
y=(−4)2=16
. If
x=3
, then
y=(3)2=9
.
Step 3: Determine Domain and Range
The domain of a function is the set of all possible input values (
x
). Since we can square any real number, the domain of
f(x)=x2
is all real numbers, denoted by
R
. The range is the set of all possible output values (
y
). Since the square of any real number is always non-negative, the range of
f(x)=x2
is all non-negative real numbers, which can be written as
[0,∞)
.
Step 4: Draw the Graph of the Function
Using the points from the completed table, we can plot them on a coordinate plane. Connecting these points with a smooth curve reveals that the graph of
f(x)=x2
is a parabola. This parabola opens upwards, is symmetric about the
y
-axis, and has its lowest point, or vertex, at the origin
(0,0)
.