Solving equations and inequalities graphically

What it shows

The solutions of f(x) = g(x) are the x-coordinates of the points where the graphs of y = f(x) and y = g(x) meet, and f(x) < g(x) holds where the graph of f lies below the graph of g. The simulation finds the intersections numerically (change of sign, then bisection), so it also solves equations such as 2^x = x + 1 that algebra cannot. Squaring both sides or multiplying by a denominator can add extraneous solutions; checking them on the graph rejects them. Only the window shown is searched. See also Sign of a quadratic function and Numerical root finding.

How to use

Choose a pair in Pair of functions or type your own f(x) and g(x); g = horizontal line y = k sets g(x) = k. Pick f = g, f < g, f ≤ g, f > g or f ≥ g, then move the k slider or drag the orange graph. Tick Plot f(x) − g(x), Solutions vs k and Algebra check, use − and + to zoom, Home window to return, and drag the background to pan.

Parameters you can change

• Starting pair of functions Parabola and line: x² − 2x − 3 and x + k, Cubic and horizontal line: x³ − 3x and k, Quartic and horizontal line: x⁴ − 4x² and k, Radical: √(x + 2) and x + k, Rational: (x² − 1)/(x − 1) and k, Absolute value: |2x − 3| and x + k, Exponential: 2^x and x + k, Custom (uses f and g below)
• Custom f(x) (e.g. x^3 - 2x)
• Custom g(x) (use k for the parameter)
• Parameter k -10–10
• Solve Equation f(x) = g(x), Inequality f(x) < g(x), Inequality f(x) ≤ g(x), Inequality f(x) > g(x), Inequality f(x) ≥ g(x)
• Plot h(x) = f(x) − g(x)
• Show the number of solutions for each k
• Check the solutions of the algebraic method

Questions to explore

1. For which values of k does x³ − 3x = k have exactly three solutions?
2. Why is x ≈ −1.618 not a solution of √(x + 2) = x + 1, although it solves the squared equation?
3. Why does (x² − 1)/(x − 1) = 2 have no solution, even though the graph looks like a straight line?