Evaluating a Composition
Evaluate composite functions f(g(x)) by substituting the inner function output into the outer: work inside-out, tracking domain restrictions of each function in the chain.
Key Concepts
Property To evaluate a composite function like $(f \circ g)(a)$, you can use two methods. Method 1: First calculate $g(a)$, then use that result as the input for $f$. Method 2: First find the general composite function $(f \circ g)(x)$, then substitute $x=a$ into the new function.
Let $f(x) = x^2+2$ and $g(x)=2x$. Find $(f \circ g)(3)$. Method 1: Find $g(3) = 2(3) = 6$. Then find $f(6) = 6^2 + 2 = 38$. So $(f \circ g)(3) = 38$. Method 2: Find $f(g(x)) = f(2x) = (2x)^2 + 2 = 4x^2 + 2$. Then plug in $x=3$: $4(3)^2 + 2 = 38$.
To solve $(f \circ g)(5)$, you can either plug 5 into $g$, get a result, and plug that result into $f$ (a two step journey). Or, you can first build the mega function $f(g(x))$ by mushing them together, and then plug 5 into your powerful new creation. Both paths lead to the same treasure at the end!
Common Questions
What does it mean to evaluate a composition of functions?
Evaluating f(g(x)) means first computing g(x) at a given input value to get an intermediate result, then using that result as the input to f. The order matters: f(g(x)) and g(f(x)) generally give different results.
How do you evaluate f(g(3)) step by step?
First substitute x=3 into g to find g(3). Then substitute that numeric result into f to find f(g(3)). For example, if g(x)=x+2 and f(x)=x^2, then g(3)=5 and f(g(3))=f(5)=25.
What domain restrictions apply when evaluating compositions?
The output of the inner function must lie within the domain of the outer function. If g produces a value outside f's domain, the composition is undefined at that input. Always verify that each intermediate result is a valid input for the next function.