Math Problem Statement
Complete the following for f(x) = 3x if 0 <= x < 2 and x + 3 if 2 <= x <= 4. (a) Determine the domain of f. (b) Evaluate f(-1), f(1), and f(5). (c) Graph f. (d) Is f continuous on its domain?
Solution
The image shows a piecewise function defined as:
3x & \text{if } 0 \leq x < 2 \\ x + 3 & \text{if } 2 \leq x \leq 4 \end{cases}$$ Several questions are asked about this function: ### (a) **Determine the domain of $$f(x)$$:** The domain of a function is the set of all $$x$$-values where the function is defined. From the piecewise function, the first part is defined for $$0 \leq x < 2$$, and the second part is defined for $$2 \leq x \leq 4$$. Combining these intervals, the domain of $$f(x)$$ is: $$[0, 4]$$ So the domain of $$f(x)$$ is $$0 \leq x \leq 4$$. ### (b) **Evaluate $$f(-1)$$, $$f(1)$$, and $$f(5)$$:** - $$f(-1)$$: Since $$-1$$ is not within the domain $$[0, 4]$$, $$f(-1)$$ is **undefined**. - $$f(1)$$: For $$x = 1$$, use the first part of the function $$3x$$, since $$1$$ falls in the interval $$0 \leq x < 2$$. So, $$f(1) = 3(1) = 3$$. - $$f(5)$$: Since $$5$$ is outside the domain $$[0, 4]$$, $$f(5)$$ is **undefined**. ### (c) **Graph $$f(x)$$:** To graph the function: - For $$0 \leq x < 2$$, plot the line $$y = 3x$$. - For $$2 \leq x \leq 4$$, plot the line $$y = x + 3$$. At $$x = 2$$, check the transition: $$f(2) = 2 + 3 = 5$$, so the graph jumps from the value $$6$$ from the line $$3x$$ to $$5$$ on the line $$x + 3$$. ### (d) **Is $$f(x)$$ continuous on its domain?** A function is continuous if there are no breaks, jumps, or gaps in the graph. Here, at $$x = 2$$: - From the first part: $$\lim_{x \to 2^-} f(x) = 3(2) = 6$$. - From the second part: $$f(2) = 2 + 3 = 5$$. Since the left-hand limit $$6$$ does not equal the value at $$x = 2$$ (which is $$5$$), the function is **not continuous** at $$x = 2$$. Therefore, $$f(x)$$ is **not continuous** on its entire domain. --- Would you like more details on any of these steps? Here are some related questions you might explore: 1. How can I check continuity for more complex piecewise functions? 2. How would I determine the range of $$f(x)$$? 3. Can you explain how to find limits at the transition points of piecewise functions? 4. What is the importance of left-hand and right-hand limits in continuity analysis? 5. How would this function behave if it were defined for all real numbers? **Tip:** When checking continuity for piecewise functions, always evaluate the limits at the boundary points of the intervals where the function changes definitions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Continuity
Limits
Formulas
f(x) = 3x (if 0 <= x < 2)
f(x) = x + 3 (if 2 <= x <= 4)
Theorems
Definition of Continuity
Piecewise Function Continuity
Suitable Grade Level
Grades 9-12
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