Certainly! Let's analyze the function [tex]\( f(x) = x^2 + 2x + 3 \)[/tex] and evaluate it at specific values.
First, we will evaluate the function at [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = (-1)^2 + 2(-1) + 3 \][/tex]
[tex]\[ = 1 - 2 + 3 \][/tex]
[tex]\[ = 2 \][/tex]
Next, we evaluate the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^2 + 2(0) + 3 \][/tex]
[tex]\[ = 0 + 0 + 3 \][/tex]
[tex]\[ = 3 \][/tex]
Now, we evaluate the function at [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1^2 + 2(1) + 3 \][/tex]
[tex]\[ = 1 + 2 + 3 \][/tex]
[tex]\[ = 6 \][/tex]
Next, we evaluate the function at [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 2^2 + 2(2) + 3 \][/tex]
[tex]\[ = 4 + 4 + 3 \][/tex]
[tex]\[ = 11 \][/tex]
Finally, we evaluate the function at [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^2 + 2(3) + 3 \][/tex]
[tex]\[ = 9 + 6 + 3 \][/tex]
[tex]\[ = 18 \][/tex]
So, the values of [tex]\( f(x) \)[/tex] for [tex]\( x = -1, 0, 1, 2, 3 \)[/tex] are:
[tex]\[ f(-1) = 2 \][/tex]
[tex]\[ f(0) = 3 \][/tex]
[tex]\[ f(1) = 6 \][/tex]
[tex]\[ f(2) = 11 \][/tex]
[tex]\[ f(3) = 18 \][/tex]
Thus, the results are [tex]\( [2, 3, 6, 11, 18] \)[/tex].
