11th Grade Q1

Composing two functions means applying one function to the output of another. Here's how you do it: To compose $f(x)$ and $g(x)$, you combine their expressions as:

Example

1. Start with the functions: Write the two functions you want to compose. For example:

1. Set up the composition: Substitute $g(x)$ into $f(x)$:

This means replacing the $x$ in $f(x)$ with $g(x)$:

1. Substitute $g(x)$ into $f(x)$: Replace $g(x)$ with its expression:

1. Simplify: Distribute and combine like terms:

1. Interpret the result: The new function $f\circ g$ is the result of applying $f$ to $g(x)$. You can now evaluate it for any $x$. For example, if $x=1$, then:

Domain of the Composition

• The domain of $f\circ g$ depends on the domain of $g(x)$ and the domain of $f(g(x))$. Ensure that $g(x)$ produces values that are valid inputs for $f$.

Summary

• Write the composition $(f\circ g)(x)=f(g(x))$.
• Replace $x$ in $f(x)$ with the expression for $g(x)$.
• Simplify the resulting expression.
• Evaluate the result for any input within the domain.