what are the domain and range of the logarithmic function $f(x) = \\log_{7}x$? use the inverse function to…
what are the domain and range of the logarithmic function $f(x) = \\log_{7}x$? use the inverse function to justify your answers.
Answer
Explanation:
Step1: Find inverse of $f(x)$
The inverse of $f(x)=\log_7 x$ is the exponential function $f^{-1}(x)=7^x$.
Step2: Identify domain of inverse
The domain of $f^{-1}(x)=7^x$ is all real numbers, so $(-\infty, \infty)$.
Step3: Link inverse domain to range
The range of $f(x)$ equals the domain of its inverse $f^{-1}(x)$.
Step4: Identify range of inverse
The range of $f^{-1}(x)=7^x$ is all positive real numbers, so $(0, \infty)$.
Step5: Link inverse range to domain
The domain of $f(x)$ equals the range of its inverse $f^{-1}(x)$.
Answer:
Domain of $f(x)=\log_7 x$: $(0, \infty)$ Range of $f(x)=\log_7 x$: $(-\infty, \infty)$