Problem 27
Question
Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ g(x, y)=\ln (4-x-y) $$
Step-by-Step Solution
Verified Answer
The region \(R\) in the \(xy\)-plane that corresponds to the domain of the function \(g(x, y) = \ln(4-x-y)\) is given by all points \((x, y)\) for which \(y < 4- x\). This represents all the area below the line \(y = 4 - x\).
1Step 1: Write down the inequality
From what has been said in the analysis, we know that the domain of a Logarithmic function is any real number that makes the expression inside the log greater than zero. So, we start with writing down the inequality for the terms inside the Logarithm: \(4-x-y > 0\).
2Step 2: Solve the inequality for y
Next, we need to solve the inequality for \(y\), essentially making \(y\) the subject. Solving it would give us: \( y<4-x\).
3Step 3: Express the domain range
Finally, the inequality \( y<4-x\) means that for any given value of \(x\), the value of \(y\) must be less than \(4 - x\). This equation describes a line in the \(xy\)-plane, and the domain that we are interested in is all the area below this line.
Key Concepts
Logarithmic InequalityDescribing Domains in CalculusInequality Solution in Two Variables
Logarithmic Inequality
A logarithmic inequality involves an expression within a logarithm that is related to a number or another expression by an inequality such as <, \( \le \), >, or \( \ge \). The key point with logarithmic inequalities, similar to logarithmic equations, is the concept that the argument of the logarithm — the expression inside the logarithm — must be positive. This is due to the fact that the logarithm function is undefined for non-positive values.
To solve a logarithmic inequality like \( \ln(4-x-y) > 0 \), we first interpret it as an inequality involving an exponent by using the properties of logarithms. The inequality tells us the set of points for \((x, y)\) for which the value inside the logarithm is positive. This approach turns the logarithmic inequality into an algebraic one, which is often more straightforward. The solution to the inequality describes a region in the coordinate plane that includes all the \((x, y)\) pairs that satisfy the initial inequality.
To solve a logarithmic inequality like \( \ln(4-x-y) > 0 \), we first interpret it as an inequality involving an exponent by using the properties of logarithms. The inequality tells us the set of points for \((x, y)\) for which the value inside the logarithm is positive. This approach turns the logarithmic inequality into an algebraic one, which is often more straightforward. The solution to the inequality describes a region in the coordinate plane that includes all the \((x, y)\) pairs that satisfy the initial inequality.
Describing Domains in Calculus
In calculus, the domain of a function is the set of all possible input values (usually \(x\)) which allows the function to work within its mathematical rules. For functions involving logarithms, as seen with \(g(x, y) = \ln(4 - x - y)\), the domain is restricted to values that make the argument of the logarithm positive. It's important to visualize and express the domain properly because it gives insight into the 'allowed' values for functions and ensures that the calculations we perform are valid.
To describe the domain of the function \(g(x, y)\), we use the inequality solution we derived, which in this case is a simple linear inequality \(y < 4 - x\). This expresses a relationship between \(x\) and \(y\) that must hold for the function to have real outputs. The domain in the xy-plane is thus all the points below the line represented by the equation \(y = 4 - x\), not including the line itself, as the function is undefined when \(4 - x - y\) equals zero.
To describe the domain of the function \(g(x, y)\), we use the inequality solution we derived, which in this case is a simple linear inequality \(y < 4 - x\). This expresses a relationship between \(x\) and \(y\) that must hold for the function to have real outputs. The domain in the xy-plane is thus all the points below the line represented by the equation \(y = 4 - x\), not including the line itself, as the function is undefined when \(4 - x - y\) equals zero.
Inequality Solution in Two Variables
Solving inequalities with two variables, like \( y < 4 - x \), requires finding the set of values that both variables can take to satisfy the inequality. Unlike a single-variable inequality, which can be represented on a number line, a two-variable inequality is represented in the coordinate plane. Here's the approach to find the solution set for our example inequality:
First, we sketch the boundary line \( y = 4 - x \) on the xy-plane. This line represents the threshold between the values that satisfy the inequality (below the line) and those that don't (on or above the line). Next, we decide if the region above or below the line fulfills the inequality - in this case, because we have a '<' sign, it's the area below the line. Hence, the solution to our inequality is every point in the plane that lies below and not on the line \( y = 4 - x \). Each point in this region translates to a pair of \(x\) and \(y\) values that, when plugged into our function, will yield a valid logarithmic output.
First, we sketch the boundary line \( y = 4 - x \) on the xy-plane. This line represents the threshold between the values that satisfy the inequality (below the line) and those that don't (on or above the line). Next, we decide if the region above or below the line fulfills the inequality - in this case, because we have a '<' sign, it's the area below the line. Hence, the solution to our inequality is every point in the plane that lies below and not on the line \( y = 4 - x \). Each point in this region translates to a pair of \(x\) and \(y\) values that, when plugged into our function, will yield a valid logarithmic output.
Other exercises in this chapter
Problem 27
Demand A store manager wants to know the demand y for an energy bar as a function of price \(x\). The daily sales for three different prices of the product are
View solution Problem 27
Evaluate \(f_{x}\) and \(f_{y}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=\ln \left(x^{2}+y^{2}\right) \quad(1,0) $$
View solution Problem 27
Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither
View solution Problem 27
Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are par
View solution