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2. The table represents some points on the graph of the quadratic function [tex]\( g \)[/tex].

[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & \frac{1}{2} & 1 & \frac{3}{2} & 2 & \frac{5}{2} & 3 \\
\hline
g(x) & \frac{7}{4} & 4 & \frac{19}{4} & 4 & \frac{7}{4} & -2 \\
\hline
\end{array}
\][/tex]

What is the range of [tex]\( g \)[/tex]?

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].

Sagot :

GinnyAnswer
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].
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