Function Composition Calculator

How it is calculated

The tool composes two straight-line functions, f(x) = a1x + b1 and g(x) = a2x + b2. Composition applies g first, then f: g(x) = a2x + b2 and (f∘g)(x) = a1 × g(x) + b1. Expanded, the composite is again a line with slope a1a2 and intercept a1b2 + b1. All five fields are required and any real number is accepted, including zero and negatives, as long as every result stays below about 9 × 10¹⁵ in size; larger values show no result. Results are rounded to two decimals. The order matters: (g∘f)(x) is a different function, with intercept a2b1 + b2, so swap the two functions to get it. If a₁ is 0, f is constant and the composite has slope 0.

Common mistakes

• Applying the functions in the wrong order. (f∘g)(x) puts x into g first and the result into f; swapping them gives intercept 1 instead of 11 for this example.
• Entering the intercept with the wrong sign. For g(x) = 4 − x, the slope is -1 and the intercept is 4, not the other way round.
• Leaving a field blank when the function has no x term. A constant function such as f(x) = 7 needs a₁ = 0 and b₁ = 7; with a blank field there is no result.
• Multiplying the two functions instead of composing them. f(x) × g(x) is a quadratic, whereas composition gives a straight line with slope a₁a₂.
• Expecting nonlinear functions to work. Only functions of the form ax + b are supported, so f(x) = x² cannot be entered.