Question

Determine whether each relation defines y as a function of x. Give the domain and range. y=√(7-2x)

Accepted Answer

Identify the given relation: $$y = \sqrt{7 - 2x}. $$This means $$y is $$defined as the square root of the expression $$7 - 2x. $$Determine the domain by finding all values of $$x$$ for which the expression inside the square root is non-negative, since the square root of a negative number is not a real number. Set up the inequality: $$7 - 2x \geq 0. $$Solve the inequality for $$x$$: Subtract 7 from both sides to get $$-2x \geq -7$$, then divide both sides by $$-2. $$Remember to reverse the inequality sign when dividing by a negative number, resulting in $$x \leq \frac{7}{2}. $$Check if the relation defines $$y as a $$function of $$x$$: For each $$x in $$the domain, there is exactly one value of $$y$$ because the square root function outputs only the non-negative root. Therefore, $$y is a $$function of $$x. $$Determine the range by considering the possible values of $$y. $$Since $$y = \sqrt{7 - 2x}$$ and the square root is always non-negative, the smallest value of $$y is 0 ($$when $$7 - 2x = 0$$), and the largest value occurs when $$7 - 2x is $$maximized (which is 7 when $$x = 0). So$$, the range is $$y \geq 0 up to$$ $$y \leq \sqrt{7}.$$

Determine whether each relation defines y as a function of x. Give the domain and range. y=√(7-2x)

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Key Concepts

Definition of a Function

Domain of a Function

Range of a Function