Answer the questions about the following function. x+6 f(x) = x-9 (a) Is the point 2, - on the graph of f? (b) If x = 1, what is f(x)? What point is on the graph of f? (c) If f(x) = 2, what is x? What point(s) is (are) on the graph of f? (d) What is the domain of f? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f.
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![**Function Analysis Exercise**
**Function:**
\[
f(x) = \frac{x + 6}{x - 9}
\]
**Questions:**
(a) Is the point \(\left( 2, -\frac{13}{2} \right)\) on the graph of \( f \)?
- Options:
- Yes, because substituting \( x = 2 \) into the given equation results in 2.
- No, because substituting \( x = 2 \) into the given equation does not result in 2.
(b) If \( x = 1 \), what is \( f(x) \)? What point is on the graph of \( f \)?
- Simplify your answer.
- Using the information in the previous step, list the point(s) on the graph of \( f \), where \( x = 1 \).
(c) If \( f(x) = 2 \), what is \( x \)? What point(s) is (are) on the graph of \( f \)?
- Simplify your answer. Use a comma to separate answers as needed.
- Using the information in the previous step, list the point(s) on the graph of \( f \), where \( f(x) = 2 \).
(d) What is the domain of \( f \)?
- The domain is \(\_\).
- Type your answer in interval notation.
(e) List the x-intercept(s), if any, of the graph of \( f \).
- Options:
- The x-intercept(s) is/are \(\_\). (Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
- There are no x-intercepts.
(f) List the y-intercept, if any, of the graph of \( f \).
- Options:
- The y-intercept(s) is/are \(\_\). (Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
- There is no y-intercept.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01f08c14-7c2f-4258-b371-70727eb7a45f%2Fa185de96-5928-4f35-b232-ebd97235418c%2Fpr4woc7_processed.png&w=3840&q=75)
![## Function Analysis and Graphing
### Given Function:
\[ f(x) = \frac{x + 6}{x - 9} \]
### Questions to Address:
1. **Point on the Graph Check:**
- Is the point \(\left(2, -\frac{13}{7}\right)\) on the graph of \( f \)?
2. **Value of the Function at a Specific Point:**
- If \( x = 1 \), calculate \( f(x) \) and find the corresponding point on the graph.
3. **Finding x for a Given y-Value:**
- If \( f(x) = 2 \), find the value of \( x \) and the corresponding points on the graph.
4. **Domain Identification:**
- Determine the domain of \( f \).
5. **x-Intercept(s):**
- List any x-intercepts on the graph.
6. **y-Intercept:**
- List the y-intercept, if it exists.
### Multiple Choice Answers and Solutions:
#### (a) Check if the Point is on the Graph:
- Choices:
- A: Yes, because substituting \( x = 2 \) into the given equation results in \(-\frac{13}{2}\).
- B: No, because substituting \( x = 2 \) into the given equation does not result in \(-\frac{13}{2}\).
- C: Yes, because substituting \( x = -\frac{13}{2} \) into the given equation results in 2.
- D: No, because substituting \( x = -\frac{13}{2} \) into the given equation does not result in 2.
#### (b) Compute \( f(x) \) for \( x = 1 \):
- Calculation: \( f(1) = \)
- Simplify your answer and find the resulting ordered pair for \( x = 1 \).
#### (c) Solve for \( f(x) = 2 \):
- Solve: \( f(x) = 2 \), produce an ordered pair or pairs as needed.
#### (d) Determine the Domain:
- Express the domain in interval notation.
#### (e) Identify x-Intercept(s):
- Indicate the x-intercept(s).
### Additional Elements:
- There are no graphs or diagrams in the provided image](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01f08c14-7c2f-4258-b371-70727eb7a45f%2Fa185de96-5928-4f35-b232-ebd97235418c%2F67e9fw_processed.png&w=3840&q=75)
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