When we say "g composed of f," we are referring to the composition of two functions $f$ and $g$, denoted as $g \circ f$. This composition means that you apply the function $f$ first and then apply the function $g$ to the result of $f$.

Mathematically, the composition $g \circ f$ is defined as: $(g \circ f)(x) = g(f(x))$ Here's a step-by-step explanation:

1. Start with a value $x$: This value is the input to the function $f$.
2. Apply the function $f$ to $x$: Calculate $f(x)$, which gives you a new value.
3. Apply the function $g$ to the result: Take the value $f(x)$ and use it as the input to the function $g$. The result is $g(f(x))$.

Example

If $f(x) = 2x + 3$ and $g(x) = x^2$, then the composition $g \circ f$ is: $(g \circ f)(x) = g(f(x)) = g(2x + 3)$ Now, apply $g(x) = x^2$ to $2x + 3$: $g(2x + 3) = (2x + 3)^2$ So, $(g \circ f)(x) = (2x + 3)^2$.

Would you like more details or have any questions?

Relative Questions:

1. How do you find the domain of a composed function?
2. What happens if the functions $f$ and $g$ are inverses?
3. Can a composition of functions be commutative, i.e., is $g \circ f = f \circ g$?
4. How does composition relate to real-world applications?
5. What are the conditions for the composition of two functions to be continuous?

Tip:

When working with function compositions, always carefully follow the order of operations, as $g \circ f$ and $f \circ g$ can yield different results.