Math Problem Statement
Graph the function f(x) = {x-3 if x <= -2, 4x+5 if x > -2}, and determine whether the function is continuous.
Solution
Let's break down the problem step-by-step:
Given function:
x - 3 & \text{if } x \leq -2, \\ 4x + 5 & \text{if } x > -2. \end{cases}$$ ### Steps to solve: 1. **Graphing the function:** - For $$x \leq -2$$, the function is linear with the equation $$f(x) = x - 3$$. This is a straight line with slope 1 and y-intercept $$-3$$. - For $$x > -2$$, the function is also linear, defined as $$f(x) = 4x + 5$$. This is a straight line with slope 4 and y-intercept 5. 2. **Determining Continuity at $$x = -2$$:** To determine if the function is continuous at $$x = -2$$, we need to check the following conditions: - **Left-hand limit**: The limit of $$f(x)$$ as $$x$$ approaches $$-2$$ from the left. - **Right-hand limit**: The limit of $$f(x)$$ as $$x$$ approaches $$-2$$ from the right. - **Value of the function at $$x = -2$$**: $$f(-2)$$. ### Calculation: 1. **Left-hand limit:** For $$x \leq -2$$, the function is $$f(x) = x - 3$$. So, as $$x \to -2^-$$, we substitute $$x = -2$$ into the expression: $$\lim_{x \to -2^-} f(x) = -2 - 3 = -5.$$ 2. **Right-hand limit:** For $$x > -2$$, the function is $$f(x) = 4x + 5$$. As $$x \to -2^+$$, we substitute $$x = -2$$ into this expression: $$\lim_{x \to -2^+} f(x) = 4(-2) + 5 = -8 + 5 = -3.$$ 3. **Value at $$x = -2$$:** Since the function at $$x = -2$$ is defined by the first piece $$f(x) = x - 3$$ for $$x \leq -2$$, we have: $$f(-2) = -2 - 3 = -5.$$ ### Conclusion: - The left-hand limit at $$x = -2$$ is $$-5$$. - The right-hand limit at $$x = -2$$ is $$-3$$. - The value of the function at $$x = -2$$ is $$-5$$. Since the left-hand limit does not equal the right-hand limit, the function **is not continuous** at $$x = -2$$. Would you like me to graph the function for better visualization? --- Here are 5 related questions to consider: 1. What is the definition of continuity at a point? 2. How do you compute limits for piecewise functions? 3. What does it mean for a function to be differentiable at a point? 4. How does the slope of a line affect its behavior as $$x$$ approaches a point? 5. What is the relationship between limits and continuity in calculus? **Tip:** When working with piecewise functions, always check both the limits and the value of the function at the boundary points to determine continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Continuity
Limits
Linear functions
Formulas
Left-hand limit: \lim_{x \to -2^-} f(x) = -5
Right-hand limit: \lim_{x \to -2^+} f(x) = -3
f(-2) = -5
Theorems
Continuity at a point
Limit theorem
Suitable Grade Level
Grades 10-12 (High School Calculus)