find the domain and range of f(x)=1/2-sin3x.

and also write the folllowing relation as the set of ordered pairs:

A relation R {0,1,2,...,10} defined by 2x+3y=12

To find the domain and range of the function \( f(x) = \frac{1}{2} - \sin(3x) \), we start by analyzing the sine function and how it affects the overall expression.

Analyzing the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of \( f(x) = \frac{1}{2} - \sin(3x) \), the sine function is defined for all real numbers. Consequently, there are no restrictions on x, which means:

• The domain of \( f(x) \) is all real numbers, expressed as:
• Domain: \( (-\infty, \infty) \)

Determining the Range

Next, let's explore the range of the function. The sine function oscillates between -1 and 1. Thus, we can find the minimum and maximum values of \( f(x) \) by substituting the extrema of the sine function into our expression:

• When \( \sin(3x) = -1 \):
• \( f(x) = \frac{1}{2} - (-1) = \frac{1}{2} + 1 = \frac{3}{2} \)
• When \( \sin(3x) = 1 \):
• \( f(x) = \frac{1}{2} - 1 = -\frac{1}{2} \)

Thus, the function oscillates between -0.5 and 1.5. Therefore, the range of the function is:

• Range: \( \left[-\frac{1}{2}, \frac{3}{2}\right] \)

Representing the Relation as Ordered Pairs

Now, let’s address the relation defined by \( R: \{0, 1, 2, \ldots, 10\} \) given by the equation \( 2x + 3y = 12 \). To express this as a set of ordered pairs, we will solve for y in terms of x:

Rearranging the equation, we get:

\( 3y = 12 - 2x \)

\( y = \frac{12 - 2x}{3} \)

Now, we can substitute each integer value from the domain \{0, 1, 2, ..., 10\} into this equation to find the corresponding y-values:

• For \( x = 0 \): \( y = \frac{12 - 2(0)}{3} = 4 \) → (0, 4)
• For \( x = 1 \): \( y = \frac{12 - 2(1)}{3} = \frac{10}{3} \) → (1, 3.33)
• For \( x = 2 \): \( y = \frac{12 - 2(2)}{3} = \frac{8}{3} \) → (2, 2.67)
• For \( x = 3 \): \( y = \frac{12 - 2(3)}{3} = 2 \) → (3, 2)
• For \( x = 4 \): \( y = \frac{12 - 2(4)}{3} = \frac{4}{3} \) → (4, 1.33)
• For \( x = 5 \): \( y = \frac{12 - 2(5)}{3} = 0 \) → (5, 0)
• For \( x = 6 \): \( y = \frac{12 - 2(6)}{3} = -\frac{4}{3} \) → (6, -1.33)
• For \( x = 7 \): \( y = \frac{12 - 2(7)}{3} = -2 \) → (7, -2)
• For \( x = 8 \): \( y = \frac{12 - 2(8)}{3} = -\frac{8}{3} \) → (8, -2.67)
• For \( x = 9 \): \( y = \frac{12 - 2(9)}{3} = -4 \) → (9, -4)
• For \( x = 10 \): \( y = \frac{12 - 2(10)}{3} = -\frac{8}{3} \) → (10, -2.67)

Thus, the relation R expressed as a set of ordered pairs is:

• R = {(0, 4), (1, 3.33), (2, 2.67), (3, 2), (4, 1.33), (5, 0), (6, -1.33), (7, -2), (8, -2.67), (9, -4), (10, -2.67)}

In summary, the domain of \( f(x) \) is all real numbers, and its range spans from -0.5 to 1.5. The relation defined by \( 2x + 3y = 12 \) when expressed as ordered pairs includes values from the domain \{0, 1, 2, ..., 10\}, with corresponding y-values calculated from the equation.