Math Problem Statement
Find the critical numbers, domain, range, x and y intercepts, table of values and it's combination of graph of this piece wise function
Solution
Analysis of the Piecewise Function :
The given function is:
\begin{cases} 4, & x < -4, \\ \frac{-2x - 4}{\sqrt{4 - x^2}}, & -4 \leq x < -2, \\ -1, & -2 \leq x \leq 2, \\ \frac{2}{\sqrt{x - 4} - 2}, & x = 3, \\ x = 4. \end{cases}$$ Let’s break this down step by step. --- ### 1. **Domain**: - For each piece of the function: - $$4$$ is valid for $$x < -4$$. - $$\frac{-2x - 4}{\sqrt{4 - x^2}}$$ is valid for $$-4 \leq x < -2$$ **provided $$4 - x^2 \geq 0$$**, which implies $$-2 \leq x \leq 2$$. - $$-1$$ is defined for $$-2 \leq x \leq 2$$. - $$\frac{2}{\sqrt{x - 4} - 2}$$ is valid for $$x = 3$$ and $$x = 4$$, though its behavior near $$x = 4$$ needs to be clarified. So the domain of $$g(x)$$ is: $$\text{Domain: } (-\infty, 2] \cup \{3, 4\}.$$ --- ### 2. **Critical Numbers**: - Critical points occur where the derivative is $$0$$ or undefined within the domain. #### For $$g(x) = 4$$: $$g(x) = 4$$ is constant, so no critical points here. #### For $$g(x) = \frac{-2x - 4}{\sqrt{4 - x^2}}$$: We compute the derivative using the quotient rule: $$g'(x) = \frac{\left(-2 \sqrt{4 - x^2}\right) - \left((-2x - 4) \cdot \frac{-x}{\sqrt{4 - x^2}}\right)}{(4 - x^2)}$$ Simplifying $$g'(x)$$: $$g'(x) = \frac{-2\sqrt{4 - x^2} - \frac{(2x + 4)(-x)}{\sqrt{4 - x^2}}}{4 - x^2}.$$ Set $$g'(x) = 0$$ to find critical points (details can be expanded if requested). #### For $$g(x) = -1$$: $$g(x) = -1$$ is constant, so no critical points. #### At $$x = 3, 4$$: These are isolated points, so no derivatives apply. --- ### 3. **Range**: The range is the set of possible output values of $$g(x)$$: $$\text{Range: } \{-1, 4\} \cup \left\{ \text{values of } \frac{-2x - 4}{\sqrt{4 - x^2}} \text{ on } [-4, -2] \right\}.$$ --- ### 4. **Intercepts**: - **x-intercepts**: Set $$g(x) = 0$$. Analyze each piece for $$g(x) = 0$$. - **y-intercepts**: Compute $$g(0)$$, if $$0$$ is within the domain. --- ### 5. **Table of Values**: Generate a table of values for $$g(x)$$ at key points (e.g., at domain boundaries and critical points). --- ### 6. **Graph Combination**: Piece together the graph using the behavior of each piece: - $$g(x) = 4$$: Horizontal line. - $$g(x) = \frac{-2x - 4}{\sqrt{4 - x^2}}$$: Analyze this fraction for asymptotes and curvature. - $$g(x) = -1$$: Horizontal line. - Points at $$x = 3$$ and $$x = 4$$. Would you like detailed computation for any specific part?Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Calculus (Derivatives)
Domain and Range Analysis
Critical Points
Graphing
Formulas
Domain: Analyze piecewise conditions and solve for valid intervals.
Derivative: Quotient Rule for functions like (-2x - 4)/sqrt(4 - x^2).
Critical Points: Derivative equals 0 or undefined.
Theorems
Piecewise Continuity
Behavior of Rational Functions near Undefined Points
Suitable Grade Level
Grades 11-12
Related Recommendation
Graph Analysis: Domain, Range, and Intercepts of a Piecewise Function
Analyzing a Piecewise Function Graph: Domain, Range, and Intervals
Analyzing Piecewise Functions: Domain, Range, and Graph Sketching
Understanding and Analyzing Piecewise Functions from Graphs
Analyzing Domain and Range of a Piecewise Function