Question

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

Accepted Answer

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 $$ x  3 $$, the expression is positive, and for $$ -1 \u003c x \u003c 3 $$, 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 $$ y  is$$ $$ (-\infty, 5] .$$

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

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Key Concepts

Domain of a Function

Range of a Function

Quadratic Expressions