Math Problem Statement
f(x) = -5, if 2 ≤ x < 3; -4, if 3 ≤ x < 4; -3, if 4 ≤ x < 5; -2, if 5 ≤ x < 6; -1, if 6 ≤ x < 7. Find (a) Domain of f, (b) Range of f, (c) f(2), (d) f(3), (e) f(7), (f) f(3.18), (g) f(sqrt(20.14))
Solution
Let's go through the given problem step by step. The piecewise function is defined as follows:
\begin{cases} -5, & \text{if } 2 \leq x < 3 \\ -4, & \text{if } 3 \leq x < 4 \\ -3, & \text{if } 4 \leq x < 5 \\ -2, & \text{if } 5 \leq x < 6 \\ -1, & \text{if } 6 \leq x < 7 \end{cases}$$ We need to solve the following parts: ### (a) Domain of $$f$$: The domain of $$f$$ is the set of all possible values of $$x$$ for which $$f(x)$$ is defined. From the piecewise definition, we see that the function is defined for $$2 \leq x < 7$$. So, the domain is: $$\boxed{[2, 7)}$$ ### (b) Range of $$f$$: The range of $$f$$ is the set of all output values. The function takes the values $$-5, -4, -3, -2, -1$$ depending on the interval of $$x$$. Thus, the range is: $$\boxed{\{-5, -4, -3, -2, -1\}}$$ ### (c) $$f(2)$$: According to the first case of the piecewise function, when $$2 \leq x < 3$$, $$f(x) = -5$$. Thus, $$f(2) = \boxed{-5}$$. ### (d) $$f(3)$$: For $$3 \leq x < 4$$, the function value is $$f(x) = -4$$. Thus, $$f(3) = \boxed{-4}$$. ### (e) $$f(7)$$: The function is defined for $$x < 7$$ but not at $$x = 7$$ (since the interval is open at 7). Thus, $$f(7)$$ is **undefined**, so the answer is: $$\boxed{\text{undefined}}$$ ### (f) $$f(3.18)$$: For $$3 \leq x < 4$$, we have $$f(x) = -4$$, and since $$3.18$$ is in this interval, $$f(3.18) = -4$$. Thus, $$f(3.18) = \boxed{-4}$$. ### (g) $$f(\sqrt{20.14})$$: First, we calculate $$\sqrt{20.14} \approx 4.49$$. This value falls in the interval $$4 \leq x < 5$$, where $$f(x) = -3$$. Thus, $$f(\sqrt{20.14}) = \boxed{-3}$$. --- Do you have any further questions, or would you like details on any step? Here are 5 related questions: 1. How can we determine whether a function is continuous at certain points? 2. What is the difference between a closed and open interval in domain definition? 3. Can a piecewise function have more than two cases? How would we represent that? 4. How do we find the range of more complex piecewise functions? 5. How can we graph a piecewise function like this? **Tip:** When analyzing piecewise functions, always pay attention to the endpoints of each interval, as they determine whether the function is defined at those specific points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Formulas
Definition of piecewise functions
Finding the domain and range of a function
Theorems
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Suitable Grade Level
Grades 9-12