Math Problem Statement
Using the function below, compute each function value. Leave all answers in exact form. If such value does not exist, enter DNE. f(x) =
5 x + 8 if x ≤ −6
3
x − 64 if −6 < x < 4 −3x2 + x − 3 if x ≥ 4 (a) f(−8) =
(b) f(−11) =
(c) f(0) =
(d) f(4) =
(e) f(−6) =
Solution
Let's determine the value of for each specified by evaluating it according to the correct piecewise function segment.
The function is given by:
\begin{cases} \frac{5}{x + 8} & \text{if } x \leq -6 \\ \frac{3}{x - 64} & \text{if } -6 < x < 4 \\ -3x^2 + x - 3 & \text{if } x \geq 4 \end{cases}$$ ### (a) $$f(-8)$$ Since $$-8 \leq -6$$, we use the first part of the function: $$f(-8) = \frac{5}{-8 + 8} = \frac{5}{0}$$ Since division by zero is undefined, $$f(-8) = \text{DNE}$$. ### (b) $$f(-11)$$ Since $$-11 \leq -6$$, we again use the first part of the function: $$f(-11) = \frac{5}{-11 + 8} = \frac{5}{-3} = -\frac{5}{3}$$ So, $$f(-11) = -\frac{5}{3}$$. ### (c) $$f(0)$$ Since $$-6 < 0 < 4$$, we use the second part of the function: $$f(0) = \frac{3}{0 - 64} = \frac{3}{-64} = -\frac{3}{64}$$ Thus, $$f(0) = -\frac{3}{64}$$. ### (d) $$f(4)$$ Since $$4 \geq 4$$, we use the third part of the function: $$f(4) = -3(4)^2 + 4 - 3$$ Calculating each term: $$= -3 \cdot 16 + 4 - 3 = -48 + 4 - 3 = -47$$ Therefore, $$f(4) = -47$$. ### (e) $$f(-6)$$ Since $$-6 \leq -6$$, we use the first part of the function: $$f(-6) = \frac{5}{-6 + 8} = \frac{5}{2}$$ So, $$f(-6) = \frac{5}{2}$$. ### Summary of Results - $$f(-8) = \text{DNE}$$ - $$f(-11) = -\frac{5}{3}$$ - $$f(0) = -\frac{3}{64}$$ - $$f(4) = -47$$ - $$f(-6) = \frac{5}{2}$$ Let me know if you'd like further explanation on any of these steps! --- Here are 5 additional questions to expand your understanding: 1. What happens if we change the intervals in a piecewise function? 2. How do we handle undefined values in piecewise functions? 3. Can you graph this piecewise function to visualize each interval? 4. How would the answers change if we shifted the boundaries of each function piece? 5. What role do restrictions in a function's domain play in defining exact values? **Tip:** Always double-check which interval an $$x$$-value falls into before calculating its function value in a piecewise function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Evaluating Functions
Undefined Values
Formulas
Piecewise function rules: f(x) is evaluated according to the segment in which x falls
Handling division: Division by zero is undefined
Theorems
Piecewise function evaluation theorem
Suitable Grade Level
Grades 9-11
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