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ماشین‌حساب تابع مرکب

این ماشین‌حساب f(g(x)) و g(f(x)) را برای دو تابع چندجمله‌ای پیدا می‌کند، آن‌ها را باز می‌کند و هر دو را در مقدار دلخواه x حساب می‌کند. مثلاً با f(x) = 2x + 3 و g(x) = x²، داریم f(g(x)) = 2x² + 3 و g(f(x)) = 4x² + 12x + 9.

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نتیجه

مقادیر را وارد کنید و برای دیدن نتیجه «محاسبه» را بزنید.

فرمول

(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.

نحوه استفاده از این ماشین‌حساب

  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.

مثال‌های حل‌شده

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.

اشتباه‌های رایج

  • 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.

پرسش‌های متداول

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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