Respuesta :

The domain of fog is equal to [-1,2].

What is the domain of a function?

⇒ The domain of a function is the set of numbers x for which the function is defined validly. If a function is defined over a range a ≤ x ≤ b, we say its domain is x∈[a,b]

What is a composite function?

⇒ A composite function of two functions combines the given two functions in the given order. i.e., for any given two functions f(x) and g(x), there can be 4 composite functions: f(g(x)) which is substituting g(x) into f(x) g(f(x)) which is substituting f(x) into g(x)

For Finding the Domain of a Composite Function

  1. Find the domain of g.
  2. Find the domain of f.
  3. Find those inputs x in the domain of g for which g(x) is in the domain of f. That is, exclude those inputs x from the domain of g for which g(x) is not in the domain of f. The resulting set is the domain of fog.

Calculation:

Given f(x)=√2-x and g(x)=x²-x

∴(fog)x=f{g(x)}=f{x²-x}

f{g(x)}=[tex]\sqrt{2-x^{2}+x[/tex]

⇒ [tex]\sqrt{-x^{2}+x+2[/tex]

⇒[tex]\sqrt{-x^{2}+2x-x+2[/tex]     ( splitting the middle term)

⇒ [tex]\sqrt{-x^{2}-x+2x+2[/tex]

⇒ [tex]\sqrt{-x(x+1)+2(x+1)[/tex]    (grouping the terms)

⇒ [tex]\sqrt{(-x+2)(x+1)}[/tex]      (taking out the common term (x+1))

The domain of a function is all the values x can take. Here x is inside the square root, x cannot take negative values as it will lead to imaginary roots. Therefore we have

⇒ (-x+2)(x+1)≥0 which implies x = 2 or x = -1 when the inequality is equal to 0.

We will check if the inequality is true in the three intervals x>2, between -1 and 2, and x taking values to the left of -1.

We will take three x values say x = -2,0 and 3. Then we have

for x = -2 we have the result as -4 which is negative. Inequality doesn't hold

for x = 0 we have the result 2 which is positive Inequality holds

for x = 3 we have the result -4 which is negative. Inequality doesn't hold.

Therefore the inequality holds in the interval [-2,3]

Therefore we have -2≤x≤3

⇒ Therefore the domain of fog is [-1,2]

Learn more about the domain here:

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