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Rekenmachine voor samengestelde functies

Deze rekenmachine bepaalt f(g(x)) en g(f(x)) voor twee veeltermfuncties, werkt ze uit en berekent beide bij een waarde van x die je kiest. Bijvoorbeeld met f(x) = 2x + 3 en g(x) = x²: f(g(x)) = 2x² + 3 en g(f(x)) = 4x² + 12x + 9.

Een getal of breuk

Resultaat

Vul waarden in en tik op Berekenen om het resultaat te zien.

Formule

(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))

Apply g first, then f: replace every x in f with g(x).

(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))

The other order. In general f(g(x)) and g(f(x)) are different.

Zo gebruik je deze rekenmachine

  1. Type f(x) and g(x) as polynomials, e.g. 2x + 3 and x^2.
  2. Choose a value of x to evaluate the composite functions at.
  3. Press Calculate to see f(g(x)) and g(f(x)) expanded, their values, and the substitution steps. To graph the function, open it in the graphing calculator.

Uitgewerkte voorbeelden

f(g(x)) and g(f(x)) for a linear and a quadratic function

f(g(x)): replace x in 2x + 3 with x², giving 2x² + 3.

g(f(x)): square 2x + 3, giving 4x² + 12x + 9.

At x = 2: g(2) = 4, so f(g(2)) = 11; f(2) = 7, so g(f(2)) = 49.

A shift inside a quadratic

f(g(x)) = (x + 1)² − 1 = x² + 2x.

g(f(x)) = (x² − 1) + 1 = x².

At x = 3: f(g(3)) = f(4) = 15 and g(f(3)) = g(8) = 9.

A cubic composed with a linear function

f(g(x)) = (x − 2)³ = x³ − 6x² + 12x − 8.

g(f(x)) = x³ − 2.

At x = 1 both are −1: (1 − 2)³ = −1 and 1³ − 2 = −1.

Veelgemaakte fouten

  • Multiplying instead of composing: f(g(x)) is not f(x)·g(x).
  • Working in the wrong order: f(g(x)) applies g first.
  • Substituting without brackets: with g(x) = x + 1, f(x) = x² gives (x + 1)², not x² + 1.
  • Assuming f(g(x)) = g(f(x)) — it usually isn’t.

Veelgestelde vragen

How do you find f of g of x?

Write f(x), then replace every x with the whole expression g(x) in brackets, and simplify. For f(x) = 2x + 3 and g(x) = x², f(g(x)) = 2x² + 3.

What is the domain of a composite function?

The x-values in the domain of g whose output g(x) is in the domain of f. For the polynomials this calculator takes, both domains are all real numbers, so f(g(x)) is defined everywhere.

Can I graph the function?

Yes — copy f(g(x)) into the Sumzo graphing calculator to plot it, see where it crosses the x-axis and read a table of values.

Which functions can I enter?

Polynomials up to degree 6, such as 3x^2 - x + 4 or (x + 1)^3. Fractions like 1/2 x work as coefficients.

Meer informatie