Arithmetic Combinations

Pre-Calculus · Axiom Academy

LESSON Arithmetic Combinations Adding, subtracting, multiplying, and dividing two functions — one output value at a time. 1. Four Ways to Combine Two Functions Given two functions f and g , we define four new functions. Each one says the same thing: go to an input x , read off f(x) and g(x) , then combine those two numbers. Watch the sum first. At each x the animation stacks the bar of height g(x) on top of the bar of height f(x) ; the combined top is f(x)+g(x) , and as x sweeps right it traces the sum curve (f+g)(x) . 2. Sum and Difference, Worked Out To get a formula for (f+g)(x) or (f-g)(x) , substitute the two expressions and combine like terms. Take f(x)=x^2+1 and g(x)=3x-2 : Example: (f+g)(x) and (f-g)(x) Geometrically, the difference (f-g)(x) is the signed gap between the two graphs: how far f sits above g at each x . The animation measures that gap with a vertical bracket and drops it onto a baseline to trace (f-g)(x) — positive where f is on top, negative where g overtakes it. 3. The Product Multiplies the Heights For the product we don't add — we multiply the two output values at each x . With f(x)=x+2 and g(x)=x , expand the product: Think of f(x) as a width and g(x) as a height: their product is the area of that rectangle. In the animation a rectangle of width f(x) and height g(x) rides along; its area is plotted as a dot that traces the product curve (fg)(x) .

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