Question

which function can be used to model the data in this table?

x	f(x)
2	0
6	2

(a) $f(x) = 3x$
(b) $f(x) = \frac{x}{2} - 1$
(c) $f(x) = x - 1$
(d) $f(x) = 2x - 1$

Explanation:

Step1: Test option a

Substitute \( x = 0 \) into \( f(x)=3x \), we get \( f(0)=3\times0 = 0\), but the table shows \( f(0)= - 1 \), so option a is wrong.

Step2: Test option b

Substitute \( x = 0 \) into \( f(x)=\frac{x}{2}-1 \), we get \( f(0)=\frac{0}{2}-1=-1 \), which matches the table. Substitute \( x = 2 \) into \( f(x)=\frac{x}{2}-1 \), we get \( f(2)=\frac{2}{2}-1 = 0\), which matches the table. Substitute \( x = 6 \) into \( f(x)=\frac{x}{2}-1 \), we get \( f(6)=\frac{6}{2}-1=3 - 1=2\), which matches the table.

Step3: Test option c (optional, since we already found a match)

Substitute \( x = 2 \) into \( f(x)=x - 1 \), we get \( f(2)=2-1 = 1\), but the table shows \( f(2)=0 \), so option c is wrong.

Step4: Test option d (optional, since we already found a match)

Substitute \( x = 2 \) into \( f(x)=2x - 1 \), we get \( f(2)=2\times2-1 = 3\), but the table shows \( f(2)=0 \), so option d is wrong.

Answer:

b. \( f(x)=\frac{x}{2}-1 \)