### Finding the Domain and Range of a Function Consider the function \( f(x) = \frac{1}{x-8} + 9 \). We will determine both the domain and the range of this function. #### Domain of \( f(x) \): The domain of a function is the set of all possible input values (x-values) that will not cause the function to be undefined. For \( f(x) = \frac{1}{x-8} + 9 \): 1. The function becomes undefined when the denominator is zero. 2. Set the denominator equal to zero and solve for \( x \): \[ x - 8 = 0 \] \[ x = 8 \] 3. Therefore, the function is undefined when \( x = 8 \). Hence, the domain of \( f(x) \) is all real numbers except \( x = 8 \). #### Range of \( f(x) \): The range of a function is the set of all possible output values (y-values). For \( f(x) = \frac{1}{x-8} + 9 \): 1. Consider the expression \( y = \frac{1}{x-8} + 9 \). 2. To find the range, we solve for \( y \): \[ y - 9 = \frac{1}{x-8} \] \[ \frac{1}{x-8} = y - 9 \] 3. The expression on the right side, \( y - 9 \), can take any value except zero. When \( y - 9 = 0 \): \[ y = 9 \] Thus, \( f(x) \) can take any real number value except 9. Therefore, the range of \( f(x) \) is all real numbers except \( y = 9 \). ### Summary - **The domain of \( f \) is:** All real numbers except 8 - **The range of \( f \) is:** All real numbers except 9 The available options in the image provide multiple choices to select the correct domain and range. The correct selections are: - **The domain of \( f \) is:** `all real numbers except 8` - **The range of \( f \) is:** `all real numbers except 9`
### Finding the Domain and Range of a Function Consider the function \( f(x) = \frac{1}{x-8} + 9 \). We will determine both the domain and the range of this function. #### Domain of \( f(x) \): The domain of a function is the set of all possible input values (x-values) that will not cause the function to be undefined. For \( f(x) = \frac{1}{x-8} + 9 \): 1. The function becomes undefined when the denominator is zero. 2. Set the denominator equal to zero and solve for \( x \): \[ x - 8 = 0 \] \[ x = 8 \] 3. Therefore, the function is undefined when \( x = 8 \). Hence, the domain of \( f(x) \) is all real numbers except \( x = 8 \). #### Range of \( f(x) \): The range of a function is the set of all possible output values (y-values). For \( f(x) = \frac{1}{x-8} + 9 \): 1. Consider the expression \( y = \frac{1}{x-8} + 9 \). 2. To find the range, we solve for \( y \): \[ y - 9 = \frac{1}{x-8} \] \[ \frac{1}{x-8} = y - 9 \] 3. The expression on the right side, \( y - 9 \), can take any value except zero. When \( y - 9 = 0 \): \[ y = 9 \] Thus, \( f(x) \) can take any real number value except 9. Therefore, the range of \( f(x) \) is all real numbers except \( y = 9 \). ### Summary - **The domain of \( f \) is:** All real numbers except 8 - **The range of \( f \) is:** All real numbers except 9 The available options in the image provide multiple choices to select the correct domain and range. The correct selections are: - **The domain of \( f \) is:** `all real numbers except 8` - **The range of \( f \) is:** `all real numbers except 9`
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Finding the Domain and Range of a Function
Consider the function \( f(x) = \frac{1}{x-8} + 9 \). We will determine both the domain and the range of this function.
#### Domain of \( f(x) \):
The domain of a function is the set of all possible input values (x-values) that will not cause the function to be undefined.
For \( f(x) = \frac{1}{x-8} + 9 \):
1. The function becomes undefined when the denominator is zero.
2. Set the denominator equal to zero and solve for \( x \):
\[ x - 8 = 0 \]
\[ x = 8 \]
3. Therefore, the function is undefined when \( x = 8 \).
Hence, the domain of \( f(x) \) is all real numbers except \( x = 8 \).
#### Range of \( f(x) \):
The range of a function is the set of all possible output values (y-values).
For \( f(x) = \frac{1}{x-8} + 9 \):
1. Consider the expression \( y = \frac{1}{x-8} + 9 \).
2. To find the range, we solve for \( y \):
\[ y - 9 = \frac{1}{x-8} \]
\[ \frac{1}{x-8} = y - 9 \]
3. The expression on the right side, \( y - 9 \), can take any value except zero. When \( y - 9 = 0 \):
\[ y = 9 \]
Thus, \( f(x) \) can take any real number value except 9. Therefore, the range of \( f(x) \) is all real numbers except \( y = 9 \).
### Summary
- **The domain of \( f \) is:** All real numbers except 8
- **The range of \( f \) is:** All real numbers except 9
The available options in the image provide multiple choices to select the correct domain and range. The correct selections are:
- **The domain of \( f \) is:** `all real numbers except 8`
- **The range of \( f \) is:** `all real numbers except 9`](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a18a320-2046-4089-b27e-0ca38db1d2b2%2F17b6d29b-f990-4294-8fba-c94699ccc30f%2F0j8ooxf.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Domain and Range of a Function
Consider the function \( f(x) = \frac{1}{x-8} + 9 \). We will determine both the domain and the range of this function.
#### Domain of \( f(x) \):
The domain of a function is the set of all possible input values (x-values) that will not cause the function to be undefined.
For \( f(x) = \frac{1}{x-8} + 9 \):
1. The function becomes undefined when the denominator is zero.
2. Set the denominator equal to zero and solve for \( x \):
\[ x - 8 = 0 \]
\[ x = 8 \]
3. Therefore, the function is undefined when \( x = 8 \).
Hence, the domain of \( f(x) \) is all real numbers except \( x = 8 \).
#### Range of \( f(x) \):
The range of a function is the set of all possible output values (y-values).
For \( f(x) = \frac{1}{x-8} + 9 \):
1. Consider the expression \( y = \frac{1}{x-8} + 9 \).
2. To find the range, we solve for \( y \):
\[ y - 9 = \frac{1}{x-8} \]
\[ \frac{1}{x-8} = y - 9 \]
3. The expression on the right side, \( y - 9 \), can take any value except zero. When \( y - 9 = 0 \):
\[ y = 9 \]
Thus, \( f(x) \) can take any real number value except 9. Therefore, the range of \( f(x) \) is all real numbers except \( y = 9 \).
### Summary
- **The domain of \( f \) is:** All real numbers except 8
- **The range of \( f \) is:** All real numbers except 9
The available options in the image provide multiple choices to select the correct domain and range. The correct selections are:
- **The domain of \( f \) is:** `all real numbers except 8`
- **The range of \( f \) is:** `all real numbers except 9`
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