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Chapter 1: Problem 49

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ g(x)=\left\\{\begin{array}{ll} 2 x-4, & x \leq 3 \\ x^{2}-2 x, & x>3 \end{array}\right. $$

Short Answer

Expert verified
The function \( g(x)=\left\{\begin{array}{ll} 2 x-4, & x \leq 3 \ x^{2}-2 x, & x>3 \end{array}\right. \) is not continuous at \(x = 3\).

Step by step solution

01

Graph the function

Using a graphing utility, graph the function \( g(x)=\left\{\begin{array}{ll} 2 x-4, & x \leq 3 \ x^{2}-2 x, & x>3 \end{array}\right. \) . Plot the two equations on the same graph: \( y = 2x - 4\) for \(x \leq 3\) and \( y = x^{2} - 2x\) for \(x > 3\).
02

Check for Continuity

Examine the graph for any points where the graph is not continuous. This would typically be demonstrated by a jump or break in the graph. In this case, the point to check for continuity is \(x = 3\). So, compare the values of the function at \(x = 3\) for both equations.
03

Determine the x-values

For the x-value at which the function is not continuous, first calculate the value of the function from both sides of the point. At \(x = 3\), from the left side, using \(y = 2x -4\), the function value is \(2*3 -4 = 2\). From right side, using \(y = x^{2} - 2x\), the function value is \(3^{2} - 2*3 = 3\). Since these values do not match, the function is not continuous at \(x = 3\).

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Most popular questions from this chapter

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