If the definition field of function f (x) = (MX ^ 2 + 4x + m + 2) ^ - 3 / 4 + (x ^ 2-mx + 1) ^ 1 / 2 is r, find the value range of real number M The answer is (√ 5) - 1 | Math enthusiast | Mathematical art

If the definition field of function f (x) = (MX ^ 2 + 4x + m + 2) ^ - 3 / 4 + (x ^ 2-mx + 1) ^ 1 / 2 is r, find the value range of real number M The answer is (√ 5) - 1

The function is meaningful and needs to meet the following conditions: MX ^ 2 + 4x + m + 2 > 0, x ^ 2-mx + 1 ≥ 0 (non negative number under root sign, denominator not 0)
From the parabola image, when x ∈ R, the inequality must be m > 0, that is, the opening is upward
It also needs to meet the following requirements: △ 1 = 4 ^ 2-4m (M + 2)

RELATED INFORMATIONS

1. If the function f (x) = MX / 4x-3 (x is not equal to 3 / 4) has f {f (x)} = x in the domain of definition, then M=
2. Given that the domain of the function f (x) = √ MX2 + 4x + 4 is a set of real numbers, the value range of the real number m is obtained
3. Given that the definition field of function f (x) = √ (m-1) x ^ 2 + 2 (m-1) x + 3 is real number set R, the value range of real number m is obtained
4. If the function f (x) is a monotone function over the domain D, and there exists an interval [a, b] &; D (where a < b), such that when x ∈ [a, b], the value range of F (x) is exactly [a, b], then the function f (x) is said to be a positive function over D, and the interval [a, b] is called an equal domain interval (1) Given that f (x) = x 12 is a positive function on [0, + ∞), find the equal domain interval of F (x); (2) Try to explore whether there is a real number m, so that the function g (x) = x2 + m is a positive function on (- ∞, 0). If there is, ask for the value range of the real number m; if not, please explain the reason
5. Let f (x) be defined as D. if there is a non-zero real number m such that for any x ∈ m (M is contained in D), there is (x-m) ∈ D and f (x-m) ≤ f (x), then F (x) is a low-key function of degree m on M. if f (x) whose domain is R is an odd function, when x ≥ 0, f (x) = | X - A ^ 2 | - A ^ 2, and f (x) is a low-key function of degree 5 on R, then the value range of real number a is?
6. Let the domain of definition of function f (x) be r, and there are two propositions as follows: 1. If there is a constant m such that any x ∈ R, there is f (x) 2. If there exists x0 ∈ R, such that for any x ∈ R, and X is not equal to x0, f (x)
7. It is known that the function f (x) is an odd function over the domain R. when x > = 0, f (x) = 2 ^ x + 2x + m (M is a constant), what is the value of F (- 1) A. - 3 B. - 1 C.1 D.3 I have to - 4-m. what about M? The answers are all constants
8. At the same time, passenger cars and freight cars depart from two places 720 km apart and meet each other in six hours. The speed ratio of passenger cars and freight cars is 7:5. How much does freight car travel per hour?
9. A truck drives from place a to place B at the speed of 30km / h. one hour after starting, a car also drives from place a to place B at the speed of 50km / h. it arrives at place B half an hour earlier than the truck. Find the distance between a and B
10. What are the general components of bridge construction
11. When m is the value, the domain of F (x) = (MX ^ 2 + 4x + m + 2) ^ - 3 / 4 + (x ^ 2-mx + 1) is r
12. When m is the value, the domain of F (x) = (MX ^ 2 + 4x + m + 2) ^ 1 / 2 + (x ^ 2-mx + 1) ^ 0 is r
13. Seeking monotonicity of function Judge the monotonicity of function f (x) = x + 4 / X (x > 0) Please write down the process, and are given
14. It is known that f (x) is an odd function defined on (- ∞, + ∞), and f (x) is a decreasing function on [0, + ∞) A. f（5）＞f（-5）B. f（4）＞f（3）C. f（-2）＞f（2）D. f（-8）=f（8）
15. Some derivative problems of mathematical function in Senior Two (1) The monotone interval of y = 1 / (x + 1) is determined by derivative (2) F (x) = (x ^ 2-3 / 2x) e ^ x increasing function interval
16. Given the function f (x) = LNX, G (x) = 1 / 2aX & # 178; + 2x, a ≠ 0. (1) if the function H (x) = f (x) - G (x) has monotone decreasing interval, find the value range of a; (2) if the function H (x) = f (x) - G (x) [1,4], find the value range of A Find the monotone interval of y = x √ (ax-x & # 178;) (a > 0)
17. Given the function f (x) = 2ax-x3, a ＞ 0, if f (x) is an increasing function on X ∈ (0,1], find the value range of A
18. If f (lgx) > F (1), then the value range of real number x is () A. （110，1）B. （0，110）∪（1，+∞）C. （110，10）D. （0，1）∪（10，+∞）
19. If the domain of definition of function f (x) is r, for any real number a B, f (a + b) = f (a) * f (b), Let f (1) = k find f (10)
20. It is known that the function f (x) whose domain is r decreases monotonically in the interval (- ∞, 5). For any real number T, f (5 + T) = f (5-T), then the following formula must be true () A. f（-1）＜f（9）＜f（13）B. f（13）＜f（9）＜f（-1）C. f（9）＜f（-1）＜f（13）D. f（13）＜f（-1）＜f（9）