Problem 35 Sketch the graph of the function... [FREE SOLUTION]

Chapter 1: Problem 35

Sketch the graph of the function. $$g(x)=\left\\{\begin{array}{ll}x+6, & x \leq-4 \\\\\frac{1}{2} x-4, & x>-4\end{array}\right.$$

Short Answer

Expert verified

The graph is a piecewise function composed of a downward sloping line defined by $ g(x) = x + 6 $ for $ x \leq -4 $, and an upward sloping line defined by $ g(x) = 0.5x - 4 $ for $ x > -4 $. The transition point at x = -4 shows a closed circle on the downward line (at the point (-4, 2)) and an open circle on the upward line (at the point (-4, -6)).

Step by step solution

Understand Piecewise Function

A piecewise function is a function that is defined by several different formulas based on different parts of the domain. In this case, we have a piecewise function that is defined by $ g(x) = x + 6 $ for $ x \leq -4 $ and $ g(x) = 0.5x - 4 $ for $ x > -4 $. So, our domain is all real numbers, but we apply different functions depending on whether the input $ x $ is less than or equal to -4, or greater than -4.

Draw the graph of the first piece

First, graph the line $ g(x) = x + 6 $ but keep in mind that this only applies for $ x \leq -4 $. This line has a slope of 1 and a y-intercept of 6. Start from the point (0, 6) and move over 1 and down 6 for every point. This part of the graph should be a descending line when moving from left to right, with a closed circle at (-4, 2) since the value at x = -4 is included in this piece.

Draw the graph of the second piece

Next, graph the line $ g(x) = 0.5x - 4 $, but remember that this only applies for $ x > -4 $. This line has a slope of 0.5 and a y-intercept of -4. Start from the point (0, -4) and move over 2 and up 1 for every point. This part of the graph should be an ascending line when moving from left to right with an open circle at (-4, -6) since x = -4 is not included in this piece.

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91影视

Sketch the graph of the function. $$g(x)=\left\\{\begin{array}{ll}x+6, & x \leq-4 \\\\\frac{1}{2} x-4, & x>-4\end{array}\right.$$

Short Answer

Step by step solution

Understand Piecewise Function

Draw the graph of the first piece

Draw the graph of the second piece

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