Math Problem Statement

State the domain of the given piecewise-defined function, using interval notation.

f(x) =

x − 2

x + 8

if −9 < x ≤ −4

x2 − 7x + 5

if x ≥ −3

The domain is

Solution

To determine the domain of the piecewise function, we need to examine each part of the function and identify all the values of xx for which the function is defined.

Given:

\begin{cases} x - 2 & \text{if } -9 < x \leq -4 \\ x^2 - 7x + 5 & \text{if } x \geq -3 \end{cases}$$ ### Step-by-Step Analysis 1. **First Piece**: $$f(x) = x - 2$$ is defined on the interval $$-9 < x \leq -4$$. This covers all values of $$x$$ between $$-9$$ (not inclusive) and $$-4$$ (inclusive). 2. **Second Piece**: $$f(x) = x^2 - 7x + 5$$ is defined for $$x \geq -3$$. This includes all values of $$x$$ from $$-3$$ to $$+\infty$$, inclusive. ### Finding the Combined Domain The function is defined on two intervals: - From $$-9$$ (not inclusive) to $$-4$$ (inclusive). - From $$-3$$ (inclusive) to $$+\infty$$. Since there is a gap between $$-4$$ and $$-3$$, these intervals do not overlap. Thus, the domain in interval notation is: $$(-9, -4] \cup [-3, \infty)$$ ### Final Answer The domain of the function is: $$(-9, -4] \cup [-3, \infty)$$ Would you like further explanation or have questions? --- Here are five related questions to explore this concept: 1. How would the domain change if the interval in the first piece were closed at $$-9$$? 2. What is the range of this piecewise function? 3. How would you determine continuity at the boundaries of each piece? 4. If a new piece were added for values between $$-4$$ and $$-3$$, how would that affect the domain? 5. How can you determine if the function is increasing or decreasing on each interval? **Tip:** When working with piecewise functions, always analyze each piece separately to ensure you understand the function's full behavior and domain.

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Math Problem Analysis

Mathematical Concepts

Algebra
Piecewise Functions
Domain of Functions

Formulas

Domain of a function
Interval notation

Theorems

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Suitable Grade Level

Grades 10-12