GraphsProve 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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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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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)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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


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


We'll factorize by x:


We'll put each factor as zero:



We'll add 1 both sides:


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



y=2*0+1, y=1

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



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).

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