What is the range of f(x)=sqrt(x-1)+4/(x^2-4) ?

The range of f(x)=sqrt(x-1)+4/(x^2-4) is f(x) = 2.05578…

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range of f(x)=sqrt(x-1)+4/(x^2-4)

range

f (x) = x - 1 + \frac{4}{x ^{2} - 4}

f (x) \leq - 1.10662 \dots or f (x) \geq 2.05578 \dots

Solution steps

Find the minimum and maximum value in each defined interval and unite the results

Domain of

x - 1 + \frac{4}{x ^{2} - 4} : 1 \leq x < 2 or x > 2

Extreme Points of

x - 1 + \frac{4}{x ^{2} - 4} :

Minimum

(1, - \frac{4}{3}),

Maximum

(1.15101 \dots, - 1.10662 \dots),

Minimum

(3.73329 \dots, 2.05578 \dots)

Find the range for the interval

1 \leq x < 2 : - \infty < f (x) \leq - 1.10662 \dots

Find the range for the interval

2 < x < \infty : 2.05578 \dots \leq f (x) < \infty

Combine the ranges of all domain intervals to obtain the function range

f (x) \leq - 1.10662 \dots or f (x) \geq 2.05578 \dots

f (x) \leq - 1.10662 \dots or f (x) \geq 2.05578 \dots

2 x - 5

(5, 6), (4, 1)

f (x) = (\frac{1}{2})^{x}

f (x) = ln (- 3 x + x^{2})

3 x + 2 y = - 4

What is the range of f(x)=sqrt(x-1)+4/(x^2-4) ?
The range of f(x)=sqrt(x-1)+4/(x^2-4) is f(x)<=-1.10662…\lor f(x)>= 2.05578…