📚 Topic Summary
In mathematics, a composite function is a function that is created by applying one function to the result of another. For example, given two functions, $f(x)$ and $g(x)$, the composite function $f(g(x))$ means you first apply the function $g$ to $x$, and then apply the function $f$ to the result. Similarly, $g(f(x))$ means you first apply the function $f$ to $x$, and then apply the function $g$ to the result. The order matters, and $f(g(x))$ is generally not the same as $g(f(x))$. Mastering composite functions is crucial for calculus and beyond.
Let's practice with some exercises to solidify your understanding!
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Composite Function |
A. The set of all possible output values of a function. |
| 2. Domain |
B. A function formed by applying one function to the result of another. |
| 3. Range |
C. The value that a function approaches as the input approaches some value. |
| 4. Function |
D. A relation where each input has only one output. |
| 5. Limit |
E. The set of all possible input values of a function. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
When evaluating $f(g(x))$, the _______ function, $g(x)$, is applied first. The output of $g(x)$ then becomes the _______ of the _______ function, $f(x)$. Understanding the _______ of each function is crucial to determine if the composite function is defined. The order of operations is essential, as $f(g(x))$ is generally _______ from $g(f(x))$.
🤔 Part C: Critical Thinking
Explain, in your own words, why the order of functions matters when creating composite functions. Provide an example to illustrate your point.