Kalkulator Fungsi Gubahan
Kalkulator ini mencari f(g(x)) dan g(f(x)) bagi dua fungsi polinomial, mengembangkannya dan menilai kedua-duanya pada nilai x pilihan anda. Contohnya, dengan f(x) = 2x + 3 dan g(x) = x², f(g(x)) = 2x² + 3 dan g(f(x)) = 4x² + 12x + 9.
Keputusan
Masukkan nilai dan tekan Kira untuk melihat keputusan.
Formula
Apply g first, then f: replace every x in f with g(x).
The other order. In general f(g(x)) and g(f(x)) are different.
Cara menggunakan kalkulator ini
- Type f(x) and g(x) as polynomials, e.g. 2x + 3 and x^2.
- Choose a value of x to evaluate the composite functions at.
- 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.
Contoh berserta jalan kerja
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.
Kesilapan biasa
- 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.
Soalan lazim
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.