Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=√(7-2x)

Verified step by step guidance

1. Identify the given relation: \(y = \sqrt{7 - 2x}\). This means \(y\) is defined as the square root of the expression \$7 - 2x$.
2. Determine the domain by finding all values of \(x\) for which the expression under 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$.
3. Solve the inequality \$7 - 2x \geq 0\( to find the domain. Rearranging gives \)-2x \geq -7\(, then dividing both sides by \)-2\( (and reversing the inequality sign) gives \)x \leq \frac{7}{2}$.
4. Check if the relation defines \(y\) as a function of \(x\). Since for each \(x\) in the domain there is exactly one non-negative value of \(y\) (the principal square root), \(y\) is a function of \(x\).
5. Determine the range by considering the possible values of \(y\). Since \(y = \sqrt{7 - 2x}\) and the expression under the root ranges from \$0\( to \)7\( (when \)x\( is at the domain endpoints), the range of \)y\( is all real numbers from \)0\( to \)\sqrt{7}$ inclusive.

Key Concepts

Definition of a Function

A function is a relation where each input x corresponds to exactly one output y. To determine if y is a function of x, check that for every x-value, there is only one y-value. This ensures the relation passes the vertical line test if graphed.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the root must be non-negative to yield real outputs, restricting the domain accordingly.

Range of a Function

The range is the set of all possible output values (y-values) of the function. For y = √(7 - 2x), since the square root produces only non-negative values, the range consists of all y ≥ 0, depending on the domain restrictions.