Graph each inequality. y≥log₂(x+1)

Verified step by step guidance

1. Identify the inequality to graph: \(y \geq \log_{2}(x+1)\). This means we are looking at all points \((x,y)\) where \(y\) is greater than or equal to the logarithm base 2 of \((x+1)\).
2. Determine the domain of the function \(y = \log_{2}(x+1)\). Since the logarithm is defined only for positive arguments, set \(x+1 > 0\), which gives \(x > -1\). So, the graph will only exist for \(x > -1\).
3. Graph the boundary curve \(y = \log_{2}(x+1)\). This is the logarithmic function shifted left by 1 unit. Plot key points such as when \(x=0\), \(y=\log_{2}(1)=0\), and when \(x=3\), \(y=\log_{2}(4)=2\), to help sketch the curve.
4. Since the inequality is \(y \geq \log_{2}(x+1)\), shade the region above or on the curve. This includes the curve itself because of the 'equal to' part.
5. Draw a solid line for the curve \(y = \log_{2}(x+1)\) to indicate that points on the curve satisfy the inequality, and shade the area above this curve for all \(x > -1\).

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. For example, y = log₂(x + 1) means y is the power to which 2 must be raised to get (x + 1). Understanding the domain and range of logarithmic functions is essential for graphing them correctly.

Graphing Inequalities

Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality. For y ≥ log₂(x + 1), you first graph y = log₂(x + 1) as a boundary curve, then shade the area above or on the curve where y-values are greater than or equal to the logarithmic function.

Domain Restrictions

The domain of y = log₂(x + 1) is x > -1 because the argument of a logarithm must be positive. Recognizing domain restrictions helps in accurately plotting the graph and understanding where the inequality is defined.