GraphsProve that the graphs of the functions f(x)=2x+1 and g(x)=x^2+x+1 have a point of intersection
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)
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.
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).