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).
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).
