Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine the domain and range of the logarithmic function [tex]\( f(x) = \log_7(x) \)[/tex], we'll analyze the properties of logarithmic functions and use the inverse function for justification.
### Domain of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Definition and Properties:
- A logarithmic function [tex]\( \log_b(x) \)[/tex] (where [tex]\( b \)[/tex] is the base and [tex]\( b > 0 \)[/tex], [tex]\( b \neq 1 \)[/tex]) is defined only for positive real numbers. This is because the logarithm represents the power to which the base must be raised to get a certain number, and raising a positive number to any real power cannot result in a non-positive number.
2. Domain:
- Therefore, for [tex]\( f(x) = \log_7(x) \)[/tex], [tex]\( x \)[/tex] must be greater than 0. In other words,
[tex]\[ \text{Domain} \: D = (0, \infty) \][/tex]
### Range of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Understanding the Logarithmic Curve:
- The logarithmic function [tex]\( \log_7(x) \)[/tex] can take any real value from negative infinity to positive infinity as [tex]\( x \)[/tex] moves from just above 0 to positive infinity.
2. Range:
- Therefore,
[tex]\[ \text{Range} \: R = (-\infty, \infty) \][/tex]
### Justification Using the Inverse Function
To justify the domain and range using the inverse function, remember that the inverse of [tex]\( f(x) = \log_7(x) \)[/tex] is the exponential function [tex]\( g(y) = 7^y \)[/tex]:
1. Inverse Function:
- The inverse function of [tex]\( f(x) = \log_7(x) \)[/tex] is given by [tex]\( g(y) = 7^y \)[/tex].
2. Domain and Range of the Inverse Function:
- For [tex]\( g(y) = 7^y \)[/tex]:
- The domain is all real numbers [tex]\( y \)[/tex], i.e., [tex]\( (-\infty, \infty) \)[/tex], because any real number can be used as an exponent.
- The range is [tex]\( (0, \infty) \)[/tex], since raising 7 (a positive number) to any real power results in a positive number greater than 0.
3. Connecting with the Original Function:
- The domain of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the range of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (0, \infty) \)[/tex].
- The range of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the domain of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (-\infty, \infty) \)[/tex].
Thus, based on this analysis, the domain and range of [tex]\( f(x) = \log_7(x) \)[/tex] are:
- Domain: [tex]\( (0, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
### Domain of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Definition and Properties:
- A logarithmic function [tex]\( \log_b(x) \)[/tex] (where [tex]\( b \)[/tex] is the base and [tex]\( b > 0 \)[/tex], [tex]\( b \neq 1 \)[/tex]) is defined only for positive real numbers. This is because the logarithm represents the power to which the base must be raised to get a certain number, and raising a positive number to any real power cannot result in a non-positive number.
2. Domain:
- Therefore, for [tex]\( f(x) = \log_7(x) \)[/tex], [tex]\( x \)[/tex] must be greater than 0. In other words,
[tex]\[ \text{Domain} \: D = (0, \infty) \][/tex]
### Range of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Understanding the Logarithmic Curve:
- The logarithmic function [tex]\( \log_7(x) \)[/tex] can take any real value from negative infinity to positive infinity as [tex]\( x \)[/tex] moves from just above 0 to positive infinity.
2. Range:
- Therefore,
[tex]\[ \text{Range} \: R = (-\infty, \infty) \][/tex]
### Justification Using the Inverse Function
To justify the domain and range using the inverse function, remember that the inverse of [tex]\( f(x) = \log_7(x) \)[/tex] is the exponential function [tex]\( g(y) = 7^y \)[/tex]:
1. Inverse Function:
- The inverse function of [tex]\( f(x) = \log_7(x) \)[/tex] is given by [tex]\( g(y) = 7^y \)[/tex].
2. Domain and Range of the Inverse Function:
- For [tex]\( g(y) = 7^y \)[/tex]:
- The domain is all real numbers [tex]\( y \)[/tex], i.e., [tex]\( (-\infty, \infty) \)[/tex], because any real number can be used as an exponent.
- The range is [tex]\( (0, \infty) \)[/tex], since raising 7 (a positive number) to any real power results in a positive number greater than 0.
3. Connecting with the Original Function:
- The domain of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the range of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (0, \infty) \)[/tex].
- The range of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the domain of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (-\infty, \infty) \)[/tex].
Thus, based on this analysis, the domain and range of [tex]\( f(x) = \log_7(x) \)[/tex] are:
- Domain: [tex]\( (0, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
