Question

Найдите область определение функции f(x)=1/√5+4x-x^2 помогите пожалуйста

Answer 1

5+4x-x²>0
x²-4x-5<0
(x+1)(x-5)<0
__+__-1  ___-___5 ___+___
Ответ: x∈(-∞;-1) и (5;+∞)

Answer 2

Finding the Domain of the Function f(x) = 1/√(5+4x-x^2)

To find the domain of the function f(x) = 1/√(5+4x-x^2), we need to determine the values of x for which the function is defined. The function will be defined as long as the expression under the square root is non-negative.

The expression under the square root, 5+4x-x^2, represents a quadratic equation. To find the domain, we need to find the values of x for which this quadratic expression is non-negative.

The quadratic expression 5+4x-x^2 can be rewritten as -x^2 + 4x + 5.

To find the domain, we need to find the values of x for which -x^2 + 4x + 5 ≥ 0.

Solving for the Domain

To solve for the domain, we can use various methods such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the domain.

The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, the quadratic equation is -x^2 + 4x + 5 = 0, so a = -1, b = 4, and c = 5.

Using the quadratic formula, we can find the solutions for x and then determine the domain based on those solutions.

Applying the Quadratic Formula

Using the quadratic formula, the solutions for x are given by:

x = (-4 ± √(4^2 - 4*(-1)*5)) / (2*(-1))

Solving this equation will give us the values of x for which the quadratic expression is non-negative, and thus the domain of the function f(x) = 1/√(5+4x-x^2).

Let's calculate the solutions for x using the quadratic formula.

x = (-4 ± √(16 + 20)) / (-2)

x = (-4 ± √36) / (-2)

x = (-4 ± 6) / (-2)

So, the solutions for x are:

x = (-4 + 6) / (-2) = 1

x = (-4 - 6) / (-2) = 5

Conclusion

The domain of the function f(x) = 1/√(5+4x-x^2) is the set of all real numbers x such that x is less than or equal to 1 and x is greater than or equal to 5.

Domain: x ≤ 1 and x ≥ 5