6. Determine the domain and the range of each relation. Use a graph to help you if necessary.
a) $y=-x+3$
b) $y=(x+1)^{2}-4$
c) $y=-3 x^{2}+1$
d) $x^{2}+y^{2}=9$
e) $y=\frac{1}{x+3}$
f) $y=\sqrt{2 x+1}$

Step 1 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function. The range of a function is the set of possible output values (usually the 'y' or the function itself), which result from the function.

Step 2 ：a) \(y=-x+3\) is a linear function, so its domain and range are all real numbers.

Step 3 ：b) \(y=(x+1)^{2}-4\) is a quadratic function, so its domain is all real numbers. The range is all numbers greater than or equal to -4, because the minimum value of a square function is 0, and we subtract 4 from it.

Step 4 ：c) \(y=-3 x^{2}+1\) is a downward-opening parabola, so its domain is all real numbers. The range is all numbers less than or equal to 1, because the maximum value of a downward-opening parabola is its y-intercept, which is 1 in this case.

Step 5 ：d) \(x^{2}+y^{2}=9\) is a circle with radius 3, so its domain and range are both all numbers between -3 and 3, inclusive.

Step 6 ：e) \(y=\frac{1}{x+3}\) is a rational function, so its domain is all real numbers except -3 (because we can't divide by zero), and its range is all real numbers (because a rational function can take any real value).

Step 7 ：f) \(y=\sqrt{2 x+1}\) is a square root function, so its domain is all numbers greater than or equal to -0.5 (because the value under the square root must be nonnegative), and its range is all nonnegative real numbers (because the square root of a number is always nonnegative).

Step 8 ：Final Answer: \[\boxed{\begin{align*} &a) \text{The domain and range of } y=-x+3 \text{ are all real numbers.} \\ &b) \text{The domain of } y=(x+1)^{2}-4 \text{ is all real numbers, and the range is all numbers greater than or equal to -4.} \\ &c) \text{The domain of } y=-3 x^{2}+1 \text{ is all real numbers, and the range is all numbers less than or equal to 1.} \\ &d) \text{The domain and range of } x^{2}+y^{2}=9 \text{ are all numbers between -3 and 3, inclusive.} \\ &e) \text{The domain of } y=\frac{1}{x+3} \text{ is all real numbers except -3, and its range is all real numbers.} \\ &f) \text{The domain of } y=\sqrt{2 x+1} \text{ is all numbers greater than or equal to -0.5, and its range is all nonnegative real numbers.} \end{align*}}\]