In Exercises 1–26, graph each inequality. y≥log_2(x+1)

Verified step by step guidance

1. Step 1: Understand the inequality y \geq \log_2(x+1). This represents the region on the graph where the y-values are greater than or equal to the logarithmic function \log_2(x+1).
2. Step 2: Identify the domain of the function \log_2(x+1). Since the logarithm is only defined for positive arguments, x+1 > 0, which implies x > -1.
3. Step 3: Graph the boundary line y = \log_2(x+1). This is the curve that separates the region where the inequality holds from where it does not. The graph of \log_2(x+1) is a logarithmic curve that passes through the point (0, 0) and approaches negative infinity as x approaches -1 from the right.
4. Step 4: Determine the region to shade. Since the inequality is y \geq \log_2(x+1), shade the region above the curve y = \log_2(x+1). This includes the curve itself because of the 'greater than or equal to' part of the inequality.
5. Step 5: Check a test point. Choose a point not on the boundary, such as (0, 1), and substitute it into the inequality to verify it satisfies y \geq \log_2(x+1). If it does, the shading is correct; if not, adjust accordingly.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as '≥' (greater than or equal to) and '≤' (less 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.

Logarithmic Functions

Logarithmic functions, such as log_2(x+1), are the inverses of exponential functions. They express the power to which a base must be raised to obtain a certain value. In this case, log_2(x+1) indicates the exponent to which 2 must be raised to yield (x+1). Familiarity with the properties and behavior of logarithmic functions is crucial for accurately graphing them.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of functions on a coordinate plane. For inequalities, it is important to determine the boundary line (in this case, y = log_2(x+1)) and then shade the appropriate region that satisfies the inequality. Mastery of these techniques allows for effective visualization of solutions and their relationships.