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Composite Function Evaluator

Evaluate f(g(x)) at a given x value step by step

Select Composition Type

Linear(Linear) Linear(Quadratic) Quadratic(Linear) Sqrt(Linear)

f(u)=au+b, g(x)=cx+d. Compute f(g(x)) at x=p

f(u)=u+,g(x)=x+,x=

Composite Function Evaluation

Step 1: Evaluate inner g(x) at the given x

Step 2: Take result u = g(x)

Step 3: Evaluate outer f(u)

Step 4: f(g(x)) = f(u) = final answer

Evaluating a composite function means working from the inside out. First compute the inner function at the given input, then use that result as the input for the outer function. Always plug carefully and check domain restrictions.

⚠ Always evaluate from inside out: g(x) first, then f(g(x)). The inner function result must be in the domain of the outer function.

What Is Composite Function Evaluation?

To evaluate f(g(x)) numerically: substitute the x value into g(x), compute g(x), then substitute that result into f(u). The process is the same regardless of function types. Domain restrictions of both functions must be satisfied.

Inside-Out

Always evaluate g(x) first, then f(g(x)). Think: apply inner, then outer to the result.

Non-Commutative

f(g(x)) != g(f(x)) in general. The order matters. f(g(2)) and g(f(2)) are usually different values.

Domain Check

x must be in domain of g. g(x) must be in domain of f. Both conditions must hold for f(g(x)) to be defined.

Multiple Layers

For three layers f(g(h(x))): evaluate h first, then g, then f. Each step feeds into the next.

Teaching Example: f(u)=2u+1, g(x)=3x-2, x=5. Step 1: g(5)=3*5-2=15-2=13. Step 2: u=13. Step 3: f(13)=2*13+1=26+1=27. f(g(5))=27.

Applications

Algebra Chain Rule Calculus Programming Data Science

Frequently Asked Questions

How to evaluate f(g(x))?▼

Work inside-out: compute g(x) first, then plug into f(u). f(g(x)) = f(g(x)). Inner then outer.

What order for composition?▼

f(g(x)): g first, then f. g(f(x)): f first, then g. Not commutative - order matters.

Evaluate at x=3?▼

Plug 3 into g(x) -> get u. Then plug u into f(u). Each step produces a number.

Empty domain?▼

If g(x) is undefined at given x, or f(u) is undefined at u=g(x), the composite is undefined.

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