Mipango

Kikokotoo cha Fungu-tendo Changamani

Kikokotoo hiki kinatafuta f(g(x)) na g(f(x)) kwa mafungu-tendo mawili ya polinomia, kinayapanua, na kukokotoa yote mawili kwa thamani ya x unayochagua. Kwa mfano, ikiwa f(x) = 2x + 3 na g(x) = x², f(g(x)) = 2x² + 3 na g(f(x)) = 4x² + 12x + 9.

Namba au sehemu

Tokeo

Weka thamani kisha bonyeza Kokotoa ili kuona tokeo.

Fomula

(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.

Jinsi ya kutumia kikokotoo hiki

  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.

Mifano iliyotatuliwa

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.

Makosa ya kawaida

  • 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.

Maswali yanayoulizwa mara kwa mara

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.

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