Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To graph the logarithmic function [tex]\(f(x)\)[/tex] and determine its domain and range, let's analyze the given table and understand the properties of logarithmic functions in general.
### Understanding the Table
The table shows pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values for the logarithmic function [tex]\(f(x)\)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline \frac{1}{125} & -3 \\ \hline \frac{1}{25} & -2 \\ \hline \frac{1}{5} & -1 \\ \hline 1 & 0 \\ \hline 5 & 1 \\ \hline 25 & 2 \\ \hline 125 & 3 \\ \hline \end{array} \][/tex]
### Domain of [tex]\(f(x)\)[/tex]
For any logarithmic function [tex]\(f(x) = \log_b(x)\)[/tex] where [tex]\(b > 1\)[/tex] (the base of the logarithm), the domain is the set of all positive real numbers, since the logarithm is only defined for positive values of [tex]\(x\)[/tex].
From the table:
- There's a correspondence for positive [tex]\(x\)[/tex] values ranging from [tex]\(\frac{1}{125}\)[/tex] to [tex]\(125\)[/tex].
Thus, the domain of [tex]\(f(x)\)[/tex] is:
[tex]\[ (0, \infty) \][/tex]
### Range of [tex]\(f(x)\)[/tex]
The range of a logarithmic function [tex]\(f(x) = \log_b(x)\)[/tex] is the set of all real numbers, since as [tex]\(x\)[/tex] increases, [tex]\(f(x)\)[/tex] increases without bound, and as [tex]\(x\)[/tex] approaches zero from the positive side, [tex]\(f(x)\)[/tex] decreases without bound.
From the table:
- [tex]\(y\)[/tex] values span from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex], representing a small section of the overall range which extends infinitely in both directions.
Thus, the range of [tex]\(f(x)\)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Plotting the Function
Using the given pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values from the table, we can plot points on a coordinate plane:
[tex]\[ \begin{align*} (\frac{1}{125}, -3), & \quad (\frac{1}{25}, -2), \\ (\frac{1}{5}, -1), & \quad (1, 0), \\ (5, 1), & \quad (25, 2), \\ (125, 3) \end{align*} \][/tex]
These points can be connected smoothly because we know logarithmic functions have a continuous curve. The graph will show an asymptotic behavior towards the y-axis (as [tex]\(x\)[/tex] approaches 0 from the positive side), passing through (1, 0), and increasing without bound.
### Final Answer
Domain:
[tex]\[ (0, \infty) \][/tex]
Range:
[tex]\[ (-\infty, \infty) \][/tex]
Explanation:
The domain of the logarithmic function [tex]\(f(x)\)[/tex] includes all positive real numbers because [tex]\( \log_b(x) \)[/tex] is defined for [tex]\(x > 0\)[/tex]. The range includes all real numbers because logarithmic functions can take on any real value as [tex]\(x\)[/tex] varies over the positive real numbers.
By plotting the points from the given table and understanding the properties of logarithmic functions, you can sketch a continuous curve that asymptotically approaches the y-axis and increases indefinitely, confirming the stated domain and range.
### Understanding the Table
The table shows pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values for the logarithmic function [tex]\(f(x)\)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline \frac{1}{125} & -3 \\ \hline \frac{1}{25} & -2 \\ \hline \frac{1}{5} & -1 \\ \hline 1 & 0 \\ \hline 5 & 1 \\ \hline 25 & 2 \\ \hline 125 & 3 \\ \hline \end{array} \][/tex]
### Domain of [tex]\(f(x)\)[/tex]
For any logarithmic function [tex]\(f(x) = \log_b(x)\)[/tex] where [tex]\(b > 1\)[/tex] (the base of the logarithm), the domain is the set of all positive real numbers, since the logarithm is only defined for positive values of [tex]\(x\)[/tex].
From the table:
- There's a correspondence for positive [tex]\(x\)[/tex] values ranging from [tex]\(\frac{1}{125}\)[/tex] to [tex]\(125\)[/tex].
Thus, the domain of [tex]\(f(x)\)[/tex] is:
[tex]\[ (0, \infty) \][/tex]
### Range of [tex]\(f(x)\)[/tex]
The range of a logarithmic function [tex]\(f(x) = \log_b(x)\)[/tex] is the set of all real numbers, since as [tex]\(x\)[/tex] increases, [tex]\(f(x)\)[/tex] increases without bound, and as [tex]\(x\)[/tex] approaches zero from the positive side, [tex]\(f(x)\)[/tex] decreases without bound.
From the table:
- [tex]\(y\)[/tex] values span from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex], representing a small section of the overall range which extends infinitely in both directions.
Thus, the range of [tex]\(f(x)\)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Plotting the Function
Using the given pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values from the table, we can plot points on a coordinate plane:
[tex]\[ \begin{align*} (\frac{1}{125}, -3), & \quad (\frac{1}{25}, -2), \\ (\frac{1}{5}, -1), & \quad (1, 0), \\ (5, 1), & \quad (25, 2), \\ (125, 3) \end{align*} \][/tex]
These points can be connected smoothly because we know logarithmic functions have a continuous curve. The graph will show an asymptotic behavior towards the y-axis (as [tex]\(x\)[/tex] approaches 0 from the positive side), passing through (1, 0), and increasing without bound.
### Final Answer
Domain:
[tex]\[ (0, \infty) \][/tex]
Range:
[tex]\[ (-\infty, \infty) \][/tex]
Explanation:
The domain of the logarithmic function [tex]\(f(x)\)[/tex] includes all positive real numbers because [tex]\( \log_b(x) \)[/tex] is defined for [tex]\(x > 0\)[/tex]. The range includes all real numbers because logarithmic functions can take on any real value as [tex]\(x\)[/tex] varies over the positive real numbers.
By plotting the points from the given table and understanding the properties of logarithmic functions, you can sketch a continuous curve that asymptotically approaches the y-axis and increases indefinitely, confirming the stated domain and range.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
