Understanding Domain and Range of the Function f(x) 1/√(x^2 - 1)

When dealing with the function f(x) 1/√(x^2 - 1), it is essential to understand the domain and range of the function. This knowledge helps us determine the set of input values (domain) for which the function is defined and the range of all output values the function can produce.

Domain of the Function

The domain of a function refers to all possible values of x for which the function is defined.

For the function f(x) 1/√(x^2 - 1), the domain is determined by the condition that the square root must be defined and the denominator must not be zero. Therefore, we must ensure that x^2 - 1 0.

Step-by-Step Calculation

Consider the expression inside the square root: x^2 - 1 0.

Solve the inequality:

x^2 - 1 (x 1)(x - 1) 0

The critical points are x -1 and x 1

Test the intervals: (-∞, -1), (-1, 1), and [1, ∞)

The inequality x^2 - 1 0 is satisfied in the intervals (-∞, -1) and [1, ∞).

Hence, the domain of the function is (-∞, -1) ∪ (1, ∞). This means the function is defined for all values of x except x -1 and x 1.

Range of the Function

The range of a function is the set of all possible output values. To find the range, we need to determine the minimum and maximum values that the function can take over its domain.

For the function f(x) 1/√(x^2 - 1), we start with the domain (-∞, -1) ∪ (1, ∞) and analyze the behavior of the function within these intervals.

Let's rewrite the function for better understanding:

Y 1/√(x^2 - 1)

Squaring both sides:

Y^2 1/(x^2 - 1)

Multiplying both sides by x^2 - 1:

Y^2(x^2 - 1) 1

Rearranging:

Y^2(x^2) - Y^2 1

Y^2(x^2) Y^2 1

x^2 (Y^2 1) / Y^2

x ±√((Y^2 1) / Y^2)

For the expression to be defined, Y ne; 0. Also, the function must be real, so x^2 - 1 0.

Therefore, the range of the function is all real numbers except 0, i.e., (-∞, 0) ∪ (0, ∞).

Conclusion

In summary, for the function f(x) 1/√(x^2 - 1): The domain is (-∞, -1) ∪ (1, ∞). The range is (-∞, 0) ∪ (0, ∞).

Understanding the domain and range helps in comprehending the behavior and limitations of the function, which is crucial for further analysis and application in various fields such as physics, engineering, and mathematics.