Question

Where do the two equations `f(x) = 3x^2+ 5 and g(x)= 4x+ 4` intersect?

Answer 1

`(1/3, 16/3) and (1,8)`

Explanation:

To figure out where the two functions intersect, we can set them equal to one another and solve for `x`. Then to get the `y` coordinate of the solution(s), we plug each `x` value back into one of the two functions (they'll both give the same output).

Let's begin by setting the functions equal to one another:

`f(x) = g(x)`

`3x^2 + 5 = 4x + 4`

Now move everything to one side.

`3x^2 - 4x + 1 = 0`

This is a factorable quadratic. Let me know if you would like me to explain how to factor it, but for now I will just go ahead and write its factored form:

`(3x-1)(x-1) = 0`

Now use the property that `ab = 0` implies that `a=0 or b=0`.

`3x - 1 = 0 or x-1 = 0`

`3x = 1 or x = 1`

`x = 1/3 or x = 1`

Finally, plug each of these back into one of the two functions to get the y-values of intersection.

`g(1/3) = 4(1/3) + 4 = 16/3`

`g(1) = 4(1) + 4 = 8`

So our two points of intersection are:

`(1/3, 16/3) and (1,8)`

Final Answer