In Exercises 1–26, graph each inequality. y≥log₂(x+1)

Verified step by step guidance

1. Identify the inequality to graph: $y\ge \log 2_{}(x+1)$. This means we want to graph the region where $y$ is greater than or equal to the logarithm base 2 of $x\; +\; 1$.
2. Determine the domain of the function $y\; =\; \backslash ,\; log\_2(x\; +\; 1)$. Since the logarithm is defined only for positive arguments, set $x\; +\; 1\; >\; 0$, which simplifies to $x\; >\; -1$. 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 $x\; =\; 3$, $y\; =\; log\_2(4)\; =\; 2$. Draw a smooth curve through these points starting from $x\; =\; -1$ (where the function approaches negative infinity).
4. Since the inequality is $y\; \ge \; log\_2(x\; +\; 1)$, shade the region above or on the curve. This includes the curve itself because of the 'greater than or equal to' sign.
5. Label the graph clearly, including the boundary curve and shaded region, and indicate that the domain is $x\; >\; -1$.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, defined as y = log_b(x), where b is the base. It is only defined for positive arguments (x > 0). Understanding the shape and domain of logarithmic functions is essential for graphing and interpreting inequalities involving logs.

Graphing Inequalities

Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality. For y ≥ f(x), the graph includes the curve y = f(x) and the area above it. Correctly identifying boundary lines and shading helps visualize solution sets.

Domain Restrictions

The domain of the function y = log_2(x+1) is x > -1 because the argument of the logarithm must be positive. Recognizing domain restrictions ensures the graph and shaded region only include valid x-values, preventing errors in plotting the inequality.