To determine the range of the function [tex]\( g(x) \)[/tex] based on the given table of values, let's systematically proceed through the solution.
Firstly, let's identify the values of [tex]\( g(x) \)[/tex] from the table:
- [tex]\( g\left(\frac{1}{2}\right) = \frac{7}{4} \)[/tex]
- [tex]\( g(1) = 4 \)[/tex]
- [tex]\( g\left(\frac{3}{2}\right) = \frac{19}{4} \)[/tex]
- [tex]\( g(2) = 4 \)[/tex]
- [tex]\( g\left(\frac{5}{2}\right) = \frac{7}{4} \)[/tex]
- [tex]\( g(3) = -2 \)[/tex]
We now collect all these values to form a list:
[tex]\[ g(x) = \left\{ \frac{7}{4}, 4, \frac{19}{4}, 4, \frac{7}{4}, -2 \right\} \][/tex]
Next, let's determine the maximum value among these:
[tex]\[ \max \left\{ \frac{7}{4}, 4, \frac{19}{4}, 4, \frac{7}{4}, -2 \right\} = \frac{19}{4} \][/tex]
As the quadratic function opens downwards (we can infer because there is a maximum point and the behavior of values around it), the range of [tex]\( g(x) \)[/tex] will include all values from the maximum value downwards.
Therefore, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[ \text{All real numbers less than or equal to} \ \frac{19}{4} \][/tex]
Given the options:
A. All real numbers less than or equal to [tex]\(\frac{3}{2}\)[/tex].
B. All real numbers less than or equal to [tex]\(\frac{19}{4}\)[/tex].
C. All real numbers greater than or equal to [tex]\(\frac{3}{2}\)[/tex].
D. All real numbers greater than or equal to [tex]\(\frac{19}{4}\)[/tex].
The correct answer is:
B. All real numbers less than or equal to [tex]\(\frac{19}{4}\)[/tex].
