Question

what are the domain and range of the logarithmic function \\(f(x) = \log_{7}x\\)? use the inverse function to justify your answers.

Explanation:

Find the inverse function

The given function is:

$$f(x) = \log_7(x)$$

To find its inverse, set \(y = \log_7(x)\) and solve for \(x\):

$$x = 7^y$$

Interchanging \(x\) and \(y\) gives the inverse function:

$$f^{-1}(x) = 7^x$$

Determine domain and range of the inverse

The inverse function is an exponential function:

$$f^{-1}(x) = 7^x$$

The domain of any exponential function \(a^x\) (where \(a > 0\)) is all real numbers:

$$\text{Domain}(f^{-1}) = (-\infty, \infty)$$

The range of this exponential function is all positive real numbers:

$$\text{Range}(f^{-1}) = (0, \infty)$$

Relate inverse properties to the original function

Using the fundamental property of inverse functions:

$$\text{Domain}(f) = \text{Range}(f^{-1})$$

$$\text{Range}(f) = \text{Domain}(f^{-1})$$

State the final domain and range

Applying the relations from the previous step:

$$\text{Domain}(f) = (0, \infty)$$

$$\text{Range}(f) = (-\infty, \infty)$$

Answer:

Domain: \((0, \infty)\)
Range: \((-\infty, \infty)\)