What does f(g(x)) mean?
Evaluate g at x, then plug that output into f.
Composite Function Calculator
For linear f(x)=ax+b and g(x)=cx+d, build f∘g and optionally evaluate at x.
Enter linear f and g
f(x) = ax + b and g(x) = cx + d. Optionally evaluate at a point.
About this calculator
How to use Composite Function Calculator, plus the formulas and logic behind the results.
Tool description
Composition f(g(x)) means apply g first, then f. For linear maps the result is another linear function.
How to use
- Enter coefficients for f(x)=ax+b.
- Enter coefficients for g(x)=cx+d.
- Enter an x value to evaluate.
- Select Calculate to see (f∘g)(x) and the simplified formula.
Formulas & reference
f(g(x)) = a(cx + d) + b = (ac)x + (ad + b)g(f(x)) = c(ax + b) + d = (ca)x + (cb + d)
Reminder
- Inside function runs first
- f(g(x)) ≠ g(f(x)) in general
- Result stays linear when both are linear
- Identity when a=1,b=0 or c=1,d=0 simplifies
Calculation logic
Forms the composed linear coefficients, evaluates at x, and also reports g(f(x)) for comparison.
Composition order matters: f∘g is generally not equal to g∘f.
Composite Function Calculator FAQ
Common questions about this calculator and how to use the results.
Is composition commutative?
No. Order usually changes the result.
Why only linear functions?
This educational tool focuses on ax+b forms for clear algebra.
What is the slope of f∘g?
The product ac of the two slopes.
Can I get g(f(x)) too?
Yes—the results include both compositions.
What if a = 0?
f is constant, so f(g(x)) is the constant b.