F(g(x))

Definition

The expression f(g(x)) represents the composition of two functions, where g(x) is the inner function that is evaluated first, and its output becomes the input for the outer function f. This concept is essential for understanding how functions can be combined to create new outputs, demonstrating the relationship between different mathematical operations and transformations.

5 Must Know Facts For Your Next Test

1. To find f(g(x)), you first evaluate g(x), and then substitute that result into f, which means you can think of it as a two-step process.
2. Function composition is not necessarily commutative, meaning that f(g(x)) is not always equal to g(f(x)).
3. The domain of the composite function f(g(x)) is determined by both the domain of g and the values that g can produce that are valid inputs for f.
4. Graphically, composing functions can be visualized by chaining transformations; for instance, if g transforms x and then f transforms g(x), you see how each function impacts the overall output.
5. If f and g are defined piecewise or have restrictions, it's important to consider those when determining the composition to avoid undefined expressions.

Review Questions

• How does evaluating f(g(x)) differ from simply finding f(x) or g(x)?
  • Evaluating f(g(x)) involves a two-step process where you first compute g(x) to get an intermediate output, and then use that result as the input for the function f. This differs from finding just f(x) or g(x), where you would only focus on one function at a time. The composition allows you to explore how functions interact with each other, leading to potentially more complex outputs than what individual functions could provide.
• Explain how to determine the domain of the composite function f(g(x)).
  • To find the domain of f(g(x)), you need to consider both the domain of g and any restrictions imposed by f. First, identify all values of x that are allowed for g(x). Then, determine which outputs from g fall within the domain of f. Any x-value that does not lead to a valid input for f after passing through g must be excluded from the domain of the composite function.
• Evaluate how changing either function in the composition f(g(x)) impacts the overall behavior and output of the composite function.
  • Changing either function in the composition alters how inputs are transformed and subsequently affects the final outputs. For instance, if you modify g(x) to introduce a new transformation or discontinuity, it can impact what values are fed into f, possibly altering its behavior significantly. Similarly, changing f might reshape how g's outputs are interpreted. Understanding these changes highlights how interconnected functions are and emphasizes the importance of each function's role in shaping the overall behavior of compositions.

Related terms

Function Composition: The process of combining two functions such that the output of one function becomes the input of another.

Inverse Function: A function that reverses the effect of another function, meaning if f(x) = y, then its inverse f^{-1}(y) = x.

Domain and Range: The set of all possible input values (domain) and output values (range) for a given function.