Math Problem Statement
If f(x) = { 2x - 5 if -2 ≤ x ≤ 5, x^3 - 5 if 5 < x ≤ 6 }, find: (a) f(0), (b) f(1), (c) f(5), and (d) f(6).
Solution
The problem is about a piecewise function , defined as:
\begin{cases} 2x - 5 & \text{if } -2 \leq x \leq 5, \\ x^3 - 5 & \text{if } 5 < x \leq 6. \end{cases}$$ You are asked to evaluate: (a) $$f(0)$$, (b) $$f(1)$$, (c) $$f(5)$$, and (d) $$f(6)$$. --- ### Step-by-step Solutions: #### (a) $$f(0)$$ Since $$0$$ is in the interval $$-2 \leq x \leq 5$$, use the formula $$f(x) = 2x - 5$$: $$f(0) = 2(0) - 5 = -5.$$ #### (b) $$f(1)$$ Since $$1$$ is also in the interval $$-2 \leq x \leq 5$$, use the formula $$f(x) = 2x - 5$$: $$f(1) = 2(1) - 5 = -3.$$ #### (c) $$f(5)$$ Since $$5$$ is in the interval $$-2 \leq x \leq 5$$, use the formula $$f(x) = 2x - 5$$: $$f(5) = 2(5) - 5 = 10 - 5 = 5.$$ #### (d) $$f(6)$$ Since $$6$$ is in the interval $$5 < x \leq 6$$, use the formula $$f(x) = x^3 - 5$$: $$f(6) = 6^3 - 5 = 216 - 5 = 211.$$ --- ### Final Answers: (a) $$f(0) = -5$$ (b) $$f(1) = -3$$ (c) $$f(5) = 5$$ (d) $$f(6) = 211$$ --- Would you like a more detailed explanation or related examples? 😊 Here are five related questions for practice: 1. What is $$f(-2)$$ for the same function? 2. Evaluate $$f(4.5)$$. 3. Find the domain and range of $$f(x)$$. 4. Determine if $$f(x)$$ is continuous at $$x = 5$$. 5. What is $$f'(x)$$ for each piece of $$f(x)$$? **Tip**: When evaluating piecewise functions, always check the interval to select the correct formula!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebraic Evaluation
Formulas
f(x) = 2x - 5
f(x) = x^3 - 5
Theorems
Piecewise function evaluation rule
Suitable Grade Level
Grades 8-10
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