To solve the system of equations given the quadratic function [tex]\( f(x) \)[/tex] and the values for the linear function [tex]\( g(x) \)[/tex], we need to determine the point(s) where [tex]\( f(x) = g(x) \)[/tex]. Let's start by examining each point individually:
We know the quadratic function:
[tex]\[ f(x) = 2x^2 + x + 4 \][/tex]
and we have the linear function values [tex]\( g(x) \)[/tex] from the table.
Let's check each pair [tex]\((x, g(x))\)[/tex] to see if it satisfies the equation [tex]\( f(x) = g(x) \)[/tex].
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 1 \][/tex]
Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = 2(-2)^2 + (-2) + 4 = 8 - 2 + 4 = 10 \][/tex]
Since [tex]\( f(-2) = 10 \)[/tex], which is not equal to [tex]\( g(-2) = 1 \)[/tex], [tex]\((-2, 1)\)[/tex] is not a solution.
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = 3 \][/tex]
Calculate [tex]\( f(-1) \)[/tex]:
[tex]\[ f(-1) = 2(-1)^2 + (-1) + 4 = 2 - 1 + 4 = 5 \][/tex]
Since [tex]\( f(-1) = 5 \)[/tex], which is not equal to [tex]\( g(-1) = 3 \)[/tex], [tex]\((-1, 3)\)[/tex] is not a solution.
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 5 \][/tex]
Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 2(0)^2 + (0) + 4 = 4 \][/tex]
Since [tex]\( f(0) = 4 \)[/tex], which is not equal to [tex]\( g(0) = 5 \)[/tex], [tex]\((0, 5)\)[/tex] is not a solution.
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 7 \][/tex]
Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 2(1)^2 + (1) + 4 = 2 + 1 + 4 = 7 \][/tex]
Since [tex]\( f(1) = 7 \)[/tex], which is equal to [tex]\( g(1) = 7 \)[/tex], [tex]\((1, 7)\)[/tex] is a solution.
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 9 \][/tex]
Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 2(2)^2 + (2) + 4 = 8 + 2 + 4 = 14 \][/tex]
Since [tex]\( f(2) = 14 \)[/tex], which is not equal to [tex]\( g(2) = 9 \)[/tex], [tex]\((2, 9)\)[/tex] is not a solution.
By checking each point, we find that the only point where the quadratic function [tex]\( f(x) \)[/tex] equals the linear function [tex]\( g(x) \)[/tex] is at [tex]\( x = 1 \)[/tex].
Thus, the solution to the system of equations is the point:
[tex]\[ (1, 7) \][/tex]
