You are given a pair of equations, one representing a supply curve and the other representing a demand curve, where p is the unit price for x items.
80 p + x - 380 = 0
and
84 p - x - 40 = 0
Identify which is the supply curve and demand curve and the appropriate domain. Put the domains in interval notation . For \infty type infinity . For more than one interval use a U to represent a "union".
Domain of the supply curve
Domain of the demand curve
To identify the supply and demand curves and their respective domains, we'll analyze the given equations:
Equation 1: 80p + x - 380 = 0
Equation 2: 84p - x - 40 = 0
In general, the supply curve represents the relationship between the quantity supplied and the price, while the demand curve represents the relationship between the quantity demanded and the price.
Looking at Equation 1, we see that the coefficient of 'x' is positive, indicating that as the unit price increases, the quantity supplied also increases. This suggests that Equation 1 represents the supply curve.
On the other hand, Equation 2 has a negative coefficient for 'x', indicating an inverse relationship between price and quantity demanded. Hence, Equation 2 represents the demand curve.
Now let's determine the domain for each curve:
Domain of the Supply Curve:
To find the domain of the supply curve, we need to evaluate the possible values for 'p' and 'x' that satisfy Equation 1. Since 'p' represents the unit price and 'x' represents the quantity supplied, realistic values for these variables should be positive. Therefore, the domain of the supply curve is:
p ≥ 0 and x ≥ 0
Domain of the Demand Curve:
Similarly, to find the domain of the demand curve, we need to assess the possible values for 'p' and 'x' that satisfy Equation 2. Again, realistic values for 'p' (unit price) and 'x' (quantity demanded) should be positive. Therefore, the domain of the demand curve is:
p ≥ 0 and x ≥ 0
To express the domains in interval notation, we can write:
Domain of the Supply Curve: [0, ∞) × [0, ∞)
Domain of the Demand Curve: [0, ∞) × [0, ∞)
Note: The '×' symbol represents a Cartesian product, meaning that any combination of valid values for 'p' and 'x' within their individual domains satisfy the equation.
To identify which equation represents the supply curve and which represents the demand curve, we need to look at the coefficients of the variables.
In the given pair of equations:
1) 80p + x - 380 = 0
2) 84p - x - 40 = 0
The equation with a positive coefficient of "x" represents the supply curve, and the equation with a negative coefficient of "x" represents the demand curve.
So, equation 1) 80p + x - 380 = 0 is the supply curve.
And equation 2) 84p - x - 40 = 0 is the demand curve.
Now, let's find the domain (possible values) for "p" for both curves.
For the supply curve, we can solve the equation for "p":
80p + x - 380 = 0
80p = 380 - x
p = (380 - x)/80
The domain of the supply curve would be the set of all values of "x" for which the expression (380 - x)/80 is defined. We need to consider any possible restrictions on "x" that would make the denominator zero.
80p + x - 380 = 0
80p = 380 - x
p = (380 - x)/80
For the supply curve, there are no restrictions on "x" or "p". Therefore, the domain of the supply curve is all real numbers, which can be represented as (-∞, ∞) in interval notation.
Now, let's find the domain for the demand curve:
84p - x - 40 = 0
84p = x + 40
p = (x + 40)/84
Similarly, for the demand curve, there are no restrictions on "x" or "p". Therefore, the domain of the demand curve is also all real numbers, which can be represented as (-∞, ∞) in interval notation.
So, the domain of the supply curve is (-∞, ∞), and the domain of the demand curve is also (-∞, ∞).