Math Problem Statement
Suppose that the function f is defined for all real numbers as follows: f(x) = { 1+x^2 if x < 1, 9x-7 if x >= 1 }. Graph the function f. Then determine whether or not the function is continuous.
Solution
The given piecewise function is:
\begin{cases} 1 + x^2 & \text{if } x < 1 \\ 9x - 7 & \text{if } x \geq 1 \end{cases}$$ ### Step 1: Graph the function 1. For $$x < 1$$, the function is $$f(x) = 1 + x^2$$, a parabola opening upwards with vertex at $$(0, 1)$$. 2. For $$x \geq 1$$, the function is $$f(x) = 9x - 7$$, a straight line with slope 9 and y-intercept -7. ### Key Points for the Graph: - For $$x = 1$$: - From the first part of the piecewise function, $$f(1^-) = 1 + 1^2 = 2$$ (as $$x$$ approaches 1 from the left). - From the second part of the piecewise function, $$f(1^+) = 9(1) - 7 = 2$$ (as $$x$$ approaches 1 from the right). Both values are the same at $$x = 1$$, so the function is continuous at $$x = 1$$. Thus, the graph will have no jump or break. ### Continuity: The function is continuous at $$x = 1$$ because: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) = 2$$ This shows that the function is continuous for all real numbers. Let me know if you'd like a graphical sketch of the function or more details on any step! --- Here are 5 related questions: 1. How do you determine if a piecewise function is continuous at a specific point? 2. What are the conditions for continuity of a function at a given point? 3. How do you differentiate a piecewise function at the point where the pieces meet? 4. What is the significance of left-hand and right-hand limits in continuity? 5. Can a function be continuous but not differentiable at a point? **Tip:** When checking continuity, always compare the left-hand limit, right-hand limit, and the function's value at the point in question!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
f(x) = 1 + x^2
f(x) = 9x - 7
Theorems
Definition of Continuity
Suitable Grade Level
High School
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