# The Algebra of Functions

A series of free Intermediate Algebra Lessons from Thinkwell's online Algebra series. Thinkwell's products and courses are being used by thousands of students at schools and universities across the globe.

Direct Variation
We often use the term direct variation to describe a form of dependence of one variable on another. An equation that makes a line and crosses the origin is a form of direct variation, where the magnitude of x increases or decreases directly as y increases or decreases. Direct variation and inverse variation are used often in science when modeling activity, such as speed or velocity.

Indirect Variation
When modeling real world situations, we often use what's called inverse or indirect variation to describe a relation between two variables. Indirect variation is a relation in which the absolute value of one variable gets smaller while the other gets larger. Indirect variation and direct variation are important concepts to understand when learning equations and interpreting graphs.

Joint and Combined Variation
In Algebra, sometimes we have functions that vary in more than one element. When this happens, we say that the functions have joint variation or combined variation. Joint variation is direct variation to more than one variable (for example, d = (r)(t)). With combined variation, we have both direct variation and indirect variation

Composition of Functions
When we put two functions together, we have something called a composition of functions. For example, the expression g(f(x)) states that we should put the "f" function into the "g" function. To do this, we simply substitute the entire inner function into each of the variables in the outer function.

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