Functions
AI Assistant

Function Intersection Calculator

Find where any two functions meet — graphically and instantly

Two functions f(x) and g(x) intersect where they have the same output — i.e., where f(x) = g(x). Graphically, these are the points where the two curves cross or touch.

This is equivalent to solving f(x) − g(x) = 0, which means finding the roots of the difference function. This tool handles any pair of functions: line meets parabola, sine meets cosine, exponential meets linear — no algebra required.

Type two functions into the chat. The AI plots both, finds all intersections in the viewport, and labels each crossing point with its (x, y) coordinates.

Graph

FAQ

How do you find where two functions intersect?
Set f(x) = g(x) and solve for x. Graphically, this is where the two curves cross. Algebraically, it's equivalent to finding the roots of h(x) = f(x) − g(x) = 0. Once you have the x values, plug them back into either function to get the y coordinates.
Can I find where a line intersects a parabola?
Yes — set line = parabola: e.g., 2x + 1 = x² becomes x² − 2x − 1 = 0, solved by the quadratic formula. There can be 0, 1, or 2 intersections. The AI will find all of them graphically and explain the algebra.
What if the functions don't intersect?
If f(x) − g(x) has no real roots in the viewport, the functions don't cross in that region. Try zooming out. Some pairs (like eˣ and x²) have limited intersections; others (like sin(x) and cos(x)) intersect infinitely many times.
How is this related to solving systems of equations?
Finding the intersection of y = f(x) and y = g(x) is exactly solving the system of equations: {y = f(x), y = g(x)}. The graphical solution is the point where the two curves meet — the x value is the solution and the y value is both functions evaluated there.
What happens at a tangent intersection?
When two curves are tangent — they touch at a point but don't cross — that point is a double root of f(x) − g(x) = 0. The difference function touches zero without changing sign. These intersections are harder to find numerically and may require zooming in.

Related Topics

Root FinderSystems of EquationsQuadratic Formula Calculator