In Exercises 39–45, graph each inequality. y ≤ 2^x

Verified step by step guidance

1. Step 1: Understand the inequality y \leq 2^x. This represents the region on the graph where the y-values are less than or equal to the values of the function y = 2^x.
2. Step 2: Begin by graphing the equation y = 2^x. This is an exponential function with a base of 2, which means it will have a curve that starts from the left and rises steeply to the right.
3. Step 3: Identify the boundary line, which is the graph of y = 2^x. Since the inequality is \leq, the boundary line will be solid, indicating that points on the line are included in the solution.
4. Step 4: Shade the region below the curve y = 2^x. This represents all the points (x, y) where y is less than or equal to 2^x.
5. Step 5: Verify by selecting a test point not on the boundary line, such as (0, 0), and check if it satisfies the inequality. Since 0 \leq 2^0, the point (0, 0) is part of the solution region, confirming the correct area is shaded.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as ≤ (less than or equal to) and ≥ (greater than or equal to) to indicate the range of possible solutions. Understanding how to interpret and graph inequalities is essential for visualizing the solutions on a coordinate plane.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In the context of the given inequality, y = 2^x represents an exponential function where the base is 2. These functions grow rapidly and are crucial for understanding the behavior of the graph as x increases.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to represent mathematical relationships visually. For inequalities, it is important to determine the boundary line (in this case, y = 2^x) and then shade the appropriate region that satisfies the inequality. Understanding how to graph both the function and the shaded area is key to solving and interpreting inequalities.