Question

(11 - 12) complete the table. 11. ( f(x) = \begin{cases} x^2 - 1, & x < 3 \\ \frac{x - 1}{3}, & x geq 3 end{cases} )
table with columns ( x ) and ( f(x) ), rows for ( x = 0, 1, 2, 3, 4 ), ( f(x) ) cells empty

Explanation:

The function is a piece - wise function: \(f(x)=

$$\begin{cases}x^{2}-1, & x < 3\\\frac{x - 1}{3},&x\geq3\end{cases}$$

\)

Step 1: For \(x = 0\)

Since \(0<3\), we use the first part of the piece - wise function \(f(x)=x^{2}-1\).
Substitute \(x = 0\) into \(x^{2}-1\): \(f(0)=0^{2}-1=- 1\)

Step 2: For \(x = 1\)

Since \(1<3\), we use \(f(x)=x^{2}-1\).
Substitute \(x = 1\) into \(x^{2}-1\): \(f(1)=1^{2}-1 = 0\)

Step 3: For \(x = 2\)

Since \(2<3\), we use \(f(x)=x^{2}-1\).
Substitute \(x = 2\) into \(x^{2}-1\): \(f(2)=2^{2}-1=4 - 1=3\)

Step 4: For \(x = 3\)

Since \(3\geq3\), we use the second part of the piece - wise function \(f(x)=\frac{x - 1}{3}\).
Substitute \(x = 3\) into \(\frac{x - 1}{3}\): \(f(3)=\frac{3-1}{3}=\frac{2}{3}\)

Step 5: For \(x = 4\)

Since \(4\geq3\), we use \(f(x)=\frac{x - 1}{3}\).
Substitute \(x = 4\) into \(\frac{x - 1}{3}\): \(f(4)=\frac{4 - 1}{3}=\frac{3}{3}=1\)

Answer:

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)