Prove that the graphs of the functions f(x)=2x+1 and g(x)=x^2+x+1

have a point of intersection

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Therefore, the y coordinate of the point verify the equation of f(x) and the equation of g(x), in the same time.

2x+1=x^2+x+1

We'll move all term to one side and we'll combine like terms:

x^2-x=0

We'll factorize by x:

x*(x-1)=0

We'll put each factor as zero:

x=0

x-1=0

We'll add 1 both sides:

x=1

Now, we'll substitute the value of x in the equation of the line, because it is much more easier to compute y.

y=2x+1

x=0

y=2*0+1, y=1

So the first pair of coordinates of intercepting point: M(0,1)

x=1

y=2*1+1=3

So the second pair of coordinates of intercepting point: N(1,3).

**The functions f and g are intercepting and their intercepting points are: M(0,1) and N(1,3). **

Two graphs have a point of intersection if the equations of the graph have a solution. The equations of the graphs given are:

f(x)=2x+1 and g(x)=x^2+x+1

Equating the two we get 2x + 1 = x^2 + x + 1

=> x^2 - x = 0

=> x(x - 1) = 0

=> x = 1 and x = 0

y = 3 , 1

**The two graphs intersect at the point (1, 3) and (0, 1)**

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