The function [tex][tex]$f(x)=\frac{1}{2} x+\frac{3}{2}$[/tex][/tex] is used to complete this table.
\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline-1 & 1 \\
\hline 0 & [tex]$\frac{3}{2}$[/tex] \\
\hline 1 & 2 \\
\hline 2 & [tex]$\frac{5}{2}$[/tex] \\
\hline
\end{tabular}

Which statements are true of the given function? Check all that apply.
- [tex]f\left(\frac{-1}{2}\right)=-2[/tex]
- [tex]f(0)=\frac{3}{2}[/tex]
- [tex]f(1)=-1[/tex]
- [tex]f(2)=1[/tex]
- [tex]f(4)=\frac{7}{2}[/tex]



Answer :

To determine the validity of each statement for the function [tex]\( f(x) = \frac{1}{2}x + \frac{3}{2} \)[/tex], let's evaluate [tex]\( f(x) \)[/tex] at the specified points.

1. Evaluating [tex]\( f\left(\frac{-1}{2}\right) \)[/tex]

[tex]\[ f\left(\frac{-1}{2}\right) = \frac{1}{2} \left( \frac{-1}{2} \right) + \frac{3}{2} = \frac{-1}{4} + \frac{3}{2} = \frac{-1}{4} + \frac{6}{4} = \frac{5}{4} \][/tex]

The statement is [tex]\( f\left(\frac{-1}{2}\right) = -2 \)[/tex], but we found [tex]\( f\left(\frac{-1}{2}\right) = \frac{5}{4} \)[/tex], so this statement is False.

2. Evaluating [tex]\( f(0) \)[/tex]

[tex]\[ f(0) = \frac{1}{2} \cdot 0 + \frac{3}{2} = \frac{3}{2} \][/tex]

The statement is [tex]\( f(0) = \frac{3}{2} \)[/tex] which matches our result, so this statement is True.

3. Evaluating [tex]\( f(1) \)[/tex]

[tex]\[ f(1) = \frac{1}{2} \cdot 1 + \frac{3}{2} = \frac{1}{2} + \frac{3}{2} = 2 \][/tex]

The statement is [tex]\( f(1) = -1 \)[/tex], but we found [tex]\( f(1) = 2 \)[/tex], so this statement is False.

4. Evaluating [tex]\( f(2) \)[/tex]

[tex]\[ f(2) = \frac{1}{2} \cdot 2 + \frac{3}{2} = 1 + \frac{3}{2} = \frac{5}{2} \][/tex]

The statement is [tex]\( f(2) = 1 \)[/tex], but we found [tex]\( f(2) = \frac{5}{2} \)[/tex], so this statement is False.

5. Evaluating [tex]\( f(4) \)[/tex]

[tex]\[ f(4) = \frac{1}{2} \cdot 4 + \frac{3}{2} = 2 + \frac{3}{2} = \frac{7}{2} \][/tex]

The statement is [tex]\( f(4) = \frac{7}{2} \)[/tex] which matches our result, so this statement is True.

Based on these evaluations, the true statements are:

- [tex]\( f(0) = \frac{3}{2} \)[/tex]
- [tex]\( f(4) = \frac{7}{2} \)[/tex]

The resulting checks are: [tex]\( f\left(\frac{-1}{2}\right) = -2 \)[/tex] (False), [tex]\( f(0) = \frac{3}{2} \)[/tex] (True), [tex]\( f(1) = -1 \)[/tex] (False), [tex]\( f(2) = 1 \)[/tex] (False), [tex]\( f(4) = \frac{7}{2} \)[/tex] (True).