Find $x_{1}$ and $x_{2}$ where $f(x)$ is discontinuous (if they exist). Suppose $x_{1}
Solution
Step 1 ：The function \(f(x)\) is defined for all real numbers. However, it is discontinuous at \(x=7\) because the function value at \(x=7\) is not equal to the limit of the function as \(x\) approaches 7. This is a removable discontinuity because the function is defined at \(x=7\) but the function value at \(x=7\) is not equal to the limit of the function as \(x\) approaches 7.
Step 2 ：Let's calculate the limit of the function as \(x\) approaches 7 and the function value at \(x=7\).
Step 3 ：\(x = 7\)
Step 4 ：\(f = 4*x + 4\)
Step 5 ：\(\text{limit}_f_{\text{at}_7} = 32\)
Step 6 ：\(f_{\text{at}_7} = 22\)
Step 7 ：The limit of the function as \(x\) approaches 7 is 32, but the function value at \(x=7\) is 22. Therefore, the function is discontinuous at \(x=7\). There are no other points of discontinuity.
Step 8 ：\(\boxed{\text{Final Answer: The function is discontinuous at } x=7. \text{This is a removable discontinuity. There are no other points of discontinuity, so } x_{2} \text{ does not exist. Therefore, } x_{1}=7 \text{ and } x_{2}=\text{DNE}.}\)
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Solution
Step 1 ：The function \(f(x)\) is defined for all real numbers. However, it is discontinuous at \(x=7\) because the function value at \(x=7\) is not equal to the limit of the function as \(x\) approaches 7. This is a removable discontinuity because the function is defined at \(x=7\) but the function value at \(x=7\) is not equal to the limit of the function as \(x\) approaches 7.
Step 2 ：Let's calculate the limit of the function as \(x\) approaches 7 and the function value at \(x=7\).
Step 3 ：\(x = 7\)
Step 4 ：\(f = 4*x + 4\)
Step 5 ：\(\text{limit}_f_{\text{at}_7} = 32\)
Step 6 ：\(f_{\text{at}_7} = 22\)
Step 7 ：The limit of the function as \(x\) approaches 7 is 32, but the function value at \(x=7\) is 22. Therefore, the function is discontinuous at \(x=7\). There are no other points of discontinuity.
Step 8 ：\(\boxed{\text{Final Answer: The function is discontinuous at } x=7. \text{This is a removable discontinuity. There are no other points of discontinuity, so } x_{2} \text{ does not exist. Therefore, } x_{1}=7 \text{ and } x_{2}=\text{DNE}.}\)
