Math Problem Statement
Sketch the graph of the given function. f(x) =
−x2 if −3 < x ≤ 0
x if 0 < x < 9 WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot
Evaluate the given expressions. f(−1) = f(0) = f(1) =
Solution
Let's start by evaluating the given piecewise function and determining , , and .
The function is defined as:
-x^2 & \text{if } -3 < x \leq 0 \\ x & \text{if } 0 < x < 9 \end{cases}$$ ### Evaluations: 1. **$$f(-1)$$:** Since $$-1$$ is in the range $$-3 < x \leq 0$$, we use the first piece of the function: $$f(-1) = -(-1)^2 = -1$$ 2. **$$f(0)$$:** Since $$0$$ is in the range $$-3 < x \leq 0$$, we use the first piece of the function: $$f(0) = -(0)^2 = 0$$ 3. **$$f(1)$$:** Since $$1$$ is in the range $$0 < x < 9$$, we use the second piece of the function: $$f(1) = 1$$ ### Final answers: - $$f(-1) = -1$$ - $$f(0) = 0$$ - $$f(1) = 1$$ Would you like a sketch of the graph of this function? Feel free to ask any follow-up questions. --- Here are 5 related questions to expand on this problem: 1. How would the graph of $$f(x)$$ change if the first piece were $$x^2$$ instead of $$-x^2$$? 2. What is the behavior of $$f(x)$$ as $$x \to -3^+$$ and $$x \to 9^-$$? 3. How would you describe the continuity of this piecewise function? 4. What would happen if the function had an additional piece for $$x \geq 9$$? 5. How can we determine whether the function is differentiable at $$x = 0$$? **Tip:** When evaluating a piecewise function, always make sure to check which part of the domain the input belongs to.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Quadratic Functions
Linear Functions
Formulas
Quadratic function: f(x) = -x^2
Linear function: f(x) = x
Theorems
Continuity and differentiability of piecewise functions
Suitable Grade Level
Grades 9-12