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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 25
Capitolo 6, Problema 25

Graph each inequality. y≥log2(x+1)

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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 \((x+1)\).
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 \(x > -1\).
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.
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.
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 \(x > -1\).

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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.
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Graphs of Logarithmic Functions

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.
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Linear Inequalities

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.
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Domain Restrictions of Composed Functions
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