In Exercises 19–32, find the (a) domain and (b) range.
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𝔂 = 5 - √ x² - 2x - 3

Guida verificata passo dopo passo

Step 1: Identify the expression under the square root, which is \( x^2 - 2x - 3 \). For the square root to be defined, this expression must be greater than or equal to zero. Set up the inequality \( x^2 - 2x - 3 \geq 0 \).

Step 2: Solve the inequality \( x^2 - 2x - 3 \geq 0 \). First, find the roots of the equation \( x^2 - 2x - 3 = 0 \) by factoring or using the quadratic formula. The factored form is \( (x - 3)(x + 1) = 0 \), giving roots \( x = 3 \) and \( x = -1 \).

Step 3: Determine the intervals where \( x^2 - 2x - 3 \geq 0 \) by testing intervals around the roots. The intervals to test are \( (-\infty, -1) \), \( (-1, 3) \), and \( (3, \infty) \).

Step 4: Analyze the sign of \( x^2 - 2x - 3 \) in each interval. For \( x < -1 \) and \( x > 3 \), the expression is positive, and for \( -1 < x < 3 \), it is negative. Thus, the domain of \( y \) is \( (-\infty, -1] \cup [3, \infty) \).

Step 5: Determine the range of \( y = 5 - \sqrt{x^2 - 2x - 3} \). Since the square root function outputs non-negative values, the maximum value of \( \sqrt{x^2 - 2x - 3} \) is 0, which occurs at the endpoints of the domain. Therefore, the range of \( y \) is \( (-\infty, 5] \).

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Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the square root must be non-negative. Thus, determining the domain often involves solving inequalities to find the valid range of x-values.

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Quadratic expressions are polynomials of degree two, typically in the form ax² + bx + c. In the context of the given function, simplifying the expression under the square root, which is a quadratic, is essential for determining both the domain and range. Understanding how to factor or complete the square can aid in this analysis.

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