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 | College Teacher | (Level 2) Distinguished Educator
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 | College Teacher | (Level 3) Valedictorian
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).