(-/12 Points] DETAILS SCALCET9 2.2.006. For the function h whose graph is given, state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.)
The Correct Answer and Explanation is :
I’m unable to access the specific graphs you’ve provided, but I can guide you through the general process of analyzing a function’s graph to determine various quantities. Here’s how you can approach the problem:
1. Function Value at a Point:
- Notation: ( h(a) )
- Interpretation: This represents the value of the function ( h ) at ( x = a ).
- How to Find: Locate ( x = a ) on the graph and determine the corresponding ( y )-coordinate.
2. Limit of the Function as ( x ) Approaches a Value:
- Notation: ( \lim_{x \to a} h(x) )
- Interpretation: This is the value that ( h(x) ) approaches as ( x ) gets arbitrarily close to ( a ).
- How to Find: Examine the behavior of the function as ( x ) approaches ( a ) from both the left and the right. If the function approaches the same value from both directions, that value is the limit.
3. Left-Hand Limit:
- Notation: ( \lim_{x \to a^-} h(x) )
- Interpretation: The value that ( h(x) ) approaches as ( x ) approaches ( a ) from the left.
- How to Find: Look at the graph as ( x ) approaches ( a ) from values less than ( a ).
4. Right-Hand Limit:
- Notation: ( \lim_{x \to a^+} h(x) )
- Interpretation: The value that ( h(x) ) approaches as ( x ) approaches ( a ) from the right.
- How to Find: Look at the graph as ( x ) approaches ( a ) from values greater than ( a ).
5. Continuity at a Point:
- Condition: The function is continuous at ( x = a ) if:
- ( h(a) ) is defined.
- ( \lim_{x \to a} h(x) ) exists.
- ( \lim_{x \to a} h(x) = h(a) ).
- How to Check: Ensure that the function value at ( a ) equals the limit as ( x ) approaches ( a ).
6. Discontinuities:
- Types:
- Jump Discontinuity: Occurs when the left-hand and right-hand limits exist but are not equal.
- Infinite Discontinuity: Occurs when the function approaches infinity as ( x ) approaches ( a ).
- Removable Discontinuity: Occurs when the limit exists but the function is not defined at ( a ).
- How to Identify: Look for breaks, holes, or asymptotes in the graph at ( x = a ).
7. Asymptotes:
- Vertical Asymptote: Occurs when ( \lim_{x \to a} h(x) = \infty ) or ( -\infty ).
- Horizontal Asymptote: Occurs when ( \lim_{x \to \infty} h(x) = L ) or ( \lim_{x \to -\infty} h(x) = L ), where ( L ) is a finite number.
- How to Identify: Vertical asymptotes are indicated by the graph approaching infinity near ( x = a ). Horizontal asymptotes are indicated by the graph leveling off as ( x ) moves towards infinity or negative infinity.
8. Behavior at Infinity:
- Notation: ( \lim_{x \to \infty} h(x) ) and ( \lim_{x \to -\infty} h(x) )
- Interpretation: These limits describe the end behavior of the function as ( x ) increases or decreases without bound.
- How to Find: Observe the graph’s behavior as ( x ) moves far to the right or left.
9. Local Extrema:
- Local Maximum: A point where the function value is higher than all nearby points.
- Local Minimum: A point where the function value is lower than all nearby points.
- How to Identify: Look for peaks (local maxima) and valleys (local minima) in the graph.
10. Concavity and Inflection Points:
- Concave Up: The graph opens upwards like a cup.
- Concave Down: The graph opens downwards like a cap.
- Inflection Point: A point where the graph changes concavity.
- How to Identify: Observe the curvature of the graph. A change from concave up to concave down (or vice versa) indicates an inflection point.
By carefully analyzing the graph using these guidelines, you can determine the values and behaviors of the function at various points. If you have specific questions or need further clarification on any of these concepts, feel free to ask!