Answer:
-1 and 4/3
Step-by-step explanation:
So first we want to find the equation for the linear function g(x):
We know that it is linear, so it will follow the equation
y = mx + b
m is the slope, and b is the y-intercept
First, find the slope:
We can see that every time the x increases by 1, the output value is increased by 2
This means that the slope will be 2
Then, find the y-intercept:
The y-intercept is when the x value is equal to 0, so in this table, when the x value is 0, the output is 7
So putting these number into the equation will give us:
y = 2x + 7
Now to find the solutions:
You have to set these equations equal to each other and solve for x
The set up should look like this:
[tex]3x^{2} + x +3 = 2x+7[/tex]
Put all of the values on one side:
[tex]3x^{2} -x-4 = 0[/tex]
And then solve for x by factoring:
[tex]3x^{2} +3x-4x-4[/tex] = 0
[tex](3x^{2} +3x)+(-4x-4)[/tex] = 0
[tex]3x(x+1)-4(x+1)[/tex] = 0
(x+1)(3x-4) = 0
Finally, to get the x values:
x + 1 = 0
x = -1
and
3x - 4 = 0
3x = 4
x = 4/3
So the answers are:
x = -1 and 4/3
The solution to the system of equations that includes quadratic function f(x) and linear function g(x) is (-1,5) and (4/3,29/3).
A polynomial function is a relation where a dependent variable is equal to a polynomial expression. A polynomial expression is an expression including numbers and variables, where variables are raised to non-negative powers.
The general form of a polynomial expression is:
a₀ + a₁x + a₂x² + a₃x³ + ... + aₙxⁿ.
The highest power to a variable is the degree of the polynomial expression.
When degree = 2, the function is a quadratic function.
When degree = 1, the function is a linear function.
The quadratic function is given to us:
f(x) = 3x² + x + 3.
We need to determine the linear equation g(x). Since it's a linear equation we use the two-point method to determine the equation.
The two-point formula is:
y-y₁ = ((y₂-y₁)/(x₂-x₁))*(x-x₁)
We take the points g(2) = 11, g(1) = 9
g(x) - g(1) = ((g(2)-g(1))/(2-1))*(x-1)
or, g(x) - 9 = ((11-9)/(2-1))*(x-1)
or, g(x) - 9 = 2(x-1)
or, g(x) = 2x - 2 + 9 = 2x + 7
∴ g(x) = 2x +7, is the linear function g(x)
We are asked to find the solution to the system of equations f(x) and g(x).
To find the solution we need to check what is the common solution to both f(x) and g(x).
For that, we equate f(x) and g(x).
3x² + x + 3 = 2x + 7
or, 3x² - x - 4 = 0
or, 3x² + 3x - 4x - 4 = 0
or, 3x(x+1) -4(x+1) = 0
or, (3x-4)(x+1) = 0
∴ Either 3x-4=0 ⇒ x = 4/3
or, x+1=0 ⇒ x = -1.
g(-1) = 5 (from the table)
f(-1) = 3(-1)² + (-1) + 3 = 3 - 1 + 3 = 5
g(4/3) = 2(4/3) + 7 = 8/3 + 21/3 = 29/3
f(4/3) = 3*(4/3)² + (4/3) + 3 = 16/3 + 4/3 + 9/3 = 29/3
∴ f(-1) = g(-1) and f(4/3) = g(4/3), so
The solution to the system of equations that includes quadratic function f(x) and linear function g(x) is (-1,5) and (4/3,29/3).
Learn more about linear and quadratic equations at
https://brainly.com/question/14075672
#SPJ2
