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The table shows the values of an even function for some inputs. $\newline$ \begin{array}{c|c} x & f(x) \\hline -4 & 2 \ -3 & \ -2 & 8 \ -1 & \ 0 & 10 \ 1 & \ 2 & \ 3 & \ 4 & 0 \ \end{array} $\newline$ Complete the table.

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Full solution

Q. The table shows the values of an even function for some inputs.

\newline

\begin{array}{c|c} x & f(x) \\hline -4 & 2 \ -3 & \ -2 & 8 \ -1 & \ 0 & 10 \ 1 & \ 2 & \ 3 & \ 4 & 0 \ \end{array}

\newline

Complete the table.

Define Even Function: An even function has the property that $f(x) = f(-x)$ for all $x$ in the domain of the function. This means that the function is symmetric with respect to the $y$ -axis. We can use this property to find the missing values in the table.
Use Symmetry Property: Since $f(x)$ is even, $f(-4)$ should be equal to $f(4)$ . The table shows $f(-4) = 2$ , so $f(4)$ must also be $2$ .
Find $f(4)$ : Similarly, $f(-3)$ should be equal to $f(3)$ . The table shows $f(-3) = 8$ , so $f(3)$ must also be $8$ .
Find $f(3)$ : Next, $f(-2)$ should be equal to $f(2)$ . The table shows $f(-2) = 10$ , so $f(2)$ must also be $10$ .
Find $f(2)$ : Finally, $f(-1)$ should be equal to $f(1)$ . The table shows $f(-1) = -1$ , so $f(1)$ must also be $-1$ .
Find $f(1)$ : The value of $f(0)$ is already given in the table as $0$ . Since $f(0) = 0$ , it confirms that the function is even because $f(0) = f(-0)$ .

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