Functions
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Function Arithmetic

Add, subtract, and multiply functions — see how operations create new shapes

Function arithmetic means combining functions with the same operations you use on numbers: addition, subtraction, and multiplication. Start with f(x) = 2x + 1 and explore what happens when you add a constant, scale it, or combine it with another linear function.

The big reveal: when you multiply two linear functions, you get a quadratic. That's where parabolas come from — they're not random shapes, they're the product of two lines.

Ask the AI anything — try "What is f(x) + g(x)?" or "Why does multiplying lines give a parabola?"

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FAQ

What is function arithmetic?
Function arithmetic means combining two functions using addition, subtraction, multiplication, or division. For example, if f(x) = 2x + 1 and g(x) = x - 1, then (f + g)(x) = 3x. You apply the operation to the outputs for each value of x.
What happens when you add two linear functions?
Adding two linear functions always gives another linear function. The slopes add and the intercepts add. For example, (2x + 1) + (x - 1) = 3x. The result is still a straight line.
Why does multiplying two linear functions give a quadratic?
Each linear function has degree 1 (the highest power of x is 1). When you multiply them, the degrees add: 1 + 1 = 2. So (2x + 1)(x - 1) = 2x² - x - 1, which is a parabola. The zeros of the parabola are exactly where each original line crosses the x-axis.
What does f(x) + c do to a graph?
Adding a constant c to a function shifts its entire graph vertically. Positive c moves it up, negative c moves it down. This is the simplest function operation — you are adding the constant function g(x) = c to f(x).

Related Topics

What is a Function?Linear FunctionsQuadratic EquationsFunction CompositionFunction TransformationsFactoring