Combining Functions

Algebra 2 · Axiom Academy

Learn how to add, subtract, multiply, and divide functions, and understand the critical role of domain restrictions in function operations. Each operation creates a new function by combining the outputs of the original functions at each input value. The key question: What is the domain of the new function? Watch what the animation does with the input: the same value of x is fed into both machines side by side. Only after each machine has produced its own output do we combine those two numbers. That is exactly what ( f + g )( x ) = f ( x ) + g ( x ) says. When we add or subtract functions, we add or subtract their outputs at each input value. Graphically, this means taking the y -values at each x and combining them. The animation shows point-by-point addition: at x = 2, f (2) = 3 and g (2) = 4, so ( f + g )(2) = 7. Notice that g 's height is not redrawn from the axis — it is lifted and stacked on top of f 's height. On the second pass the same height is flipped downward instead, and that traces the difference: ( f − g )(2) = 3 − 4 = −1. Function multiplication also works point by point: multiply the outputs at each input. At x = 3: f (3) = 3 and g (3) = 1, so ( f · g )(3) = 3 · 1 = 3. Notice how multiplication can produce zeros where one function equals zero.

This is the written version of the interactive lesson above. See the full Algebra 2 course.