Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, graph the inequality. $$ \begin{array}{l} y \leq-\log (x) \\ y \leq e^{x} \end{array} $$

Step-by-Step Solution

Answer

Shade below both curves and find the overlapping region.

Step 1: Understand the Given Inequalities

We are provided with two inequalities: \( y \leq -\log(x) \) and \( y \leq e^x \). These represent regions below their respective curves when graphed.

Step 2: Graph the First Inequality

Graph the line \( y = -\log(x) \). It is a decreasing curve that approaches infinity as \( x \) approaches 0 from the right and approaches \( y = 0 \) as \( x \) increases. Shade the region below this line to satisfy \( y \leq -\log(x) \).

Step 3: Graph the Second Inequality

Graph the line \( y = e^x \). It is an increasing curve that passes through the point (0,1) and rises steeply as \( x \) increases. Shade the region below this line to satisfy \( y \leq e^x \).

Step 4: Identify the Intersection of Shaded Regions

The solution to the system of inequalities is the intersection of the two shaded regions, which represents the set of all points that satisfy both \( y \leq -\log(x) \) and \( y \leq e^x \) simultaneously.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are incredibly useful in solving equations where the variable appears as the exponent, but they also have unique graphing properties. Consider the function \( y = -\log(x) \). This graph is a reflection of the standard logarithmic curve across the x-axis due to the negative sign. The graph is naturally decreasing since as \( x \) approaches zero from the positive side, \( y \) approaches infinity. Conversely, as \( x \) increases, \( y \) approaches zero. Logarithmic functions make it easier to tackle complex equations by transforming multiplication into addition, which is necessary for solving some types of inequalities. Additionally, knowing how to interpret these graphs helps identify constraints and feasible regions in problems involving inequalities.

Exponential Functions

Exponential functions, like \( y = e^x \), are indispensable in representing continual growth or decay. The base \( e \) is a mathematical constant approximately equal to 2.71828, known for natural exponential growth processes. This function's graph is always increasing, steepening as it moves further to the right. The exponential function crosses the y-axis at \( (0,1) \), implying that when the exponent is zero, the function equals one (since any number to the zero power equals one). Due to their rapid growth rate, they are perfect for modeling real-life phenomena like population growth, radioactive decay, and interest calculations. When graphing inequalities involving exponential functions, you look for regions below the curve (as in our inequality \( y \leq e^x \)) to identify where the solutions lie.

Intersection of Inequalities

Identifying the intersection of inequalities can often be the solution to complex mathematical problems. When dealing with two inequalities, such as \( y \leq -\log(x) \) and \( y \leq e^x \), the intersection represents the set of points that satisfy both conditions. Essentially, it is the common area shaded when both inequalities are graphed on the same axes. This intersection is crucial because it narrows down the possible solutions to only those that fulfill all given constraints. In graphical terms, the solution of a system of inequalities is where these shaded regions overlap. This overlapped region gives us the subset of x-values for which the inequality conditions hold true, helping us visualize the potential of each constraint and providing a clear picture of feasible solutions.