Let's analyze the function [tex]\( f(x) = \frac{1}{2} x + \frac{3}{2} \)[/tex] and evaluate the provided statements step-by-step:
1. Statement 1: [tex]\( f\left(\frac{-1}{2}\right) = -2 \)[/tex]
- Calculate [tex]\( f\left(\frac{-1}{2}\right) \)[/tex]:
[tex]\[ f\left(\frac{-1}{2}\right) = \frac{1}{2} \left(\frac{-1}{2}\right) + \frac{3}{2} \][/tex]
Simplify:
[tex]\[ f\left(\frac{-1}{2}\right) = \frac{-1}{4} + \frac{3}{2} \][/tex]
Convert [tex]\(\frac{3}{2}\)[/tex] to the common denominator:
[tex]\[ \frac{-1}{4} + \frac{6}{4} = \frac{5}{4} = 1.25 \][/tex]
Therefore, [tex]\( f\left(\frac{-1}{2}\right) = 1.25 \)[/tex]. This statement is False.
2. Statement 2: [tex]\( f(0) = \frac{3}{2} \)[/tex]
- Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = \frac{1}{2} (0) + \frac{3}{2} \][/tex]
[tex]\[ f(0) = 0 + \frac{3}{2} = \frac{3}{2} \][/tex]
Therefore, [tex]\( f(0) = \frac{3}{2} \)[/tex]. This statement is True.
3. Statement 3: [tex]\( f(1) = -1 \)[/tex]
- Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = \frac{1}{2} (1) + \frac{3}{2} \][/tex]
[tex]\[ f(1) = \frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2 \][/tex]
Therefore, [tex]\( f(1) = 2 \)[/tex]. This statement is False.
4. Statement 4: [tex]\( f(2) = 1 \)[/tex]
- Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = \frac{1}{2} (2) + \frac{3}{2} \][/tex]
[tex]\[ f(2) = 1 + \frac{3}{2} = \frac{5}{2} = 2.5 \][/tex]
Therefore, [tex]\( f(2) = 2.5 \)[/tex]. This statement is False.
5. Statement 5: [tex]\( f(4) = \frac{7}{2} \)[/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = \frac{1}{2} (4) + \frac{3}{2} \][/tex]
[tex]\[ f(4) = 2 + \frac{3}{2} = \frac{4}{2} + \frac{3}{2} = \frac{7}{2} \][/tex]
Therefore, [tex]\( f(4) = \frac{7}{2} \)[/tex]. This statement is True.
To summarize, the true statements are:
- [tex]\( f(0) = \frac{3}{2} \)[/tex]
- [tex]\( f(4) = \frac{7}{2} \)[/tex]
