list \( \left\lvert\, \begin{array}{l}\text { The function } f(x)=\frac{3}{x} \text { is one-to-one. } \\ \text { (a) Find the inverse of } f \text { and check the answer. } \\ \text { (b) Find the domain and the range of } f \text { and } f^{-1} \\ \text { (c) Graph } f, f^{-1} \text {, and } y=x \text { on the same coordinate axes. } \\ \text { on } 6 \\ \text { ion } 7 \\ \text { tion } 8\end{array} \begin{array}{l}\text { (a) } f^{-1}(x)=\square \\ \text { (Simplify your answer. Use integers or fractions for any } \\ \text { numbers in the expression.) }\end{array}\right. \)

To find the inverse of the function \( f(x) = \frac{3}{x} \), we start by rewriting the function as \( y = \frac{3}{x} \). Next, we swap \( x \) and \( y \) to find the inverse: 1. Swap \( x \) and \( y \): \( x = \frac{3}{y} \) 2. Solve for \( y \): Multiply both sides by \( y \) to get \( xy = 3 \), hence \( y = \frac{3}{x} \). Now we have the inverse function: \( f^{-1}(x) = \frac{3}{x} \), which is the same as the original function. This means that the function is indeed its own inverse, so it checks out! Moving on to the domain and range, the function \( f(x) = \frac{3}{x} \) is defined for all \( x \) except \( 0 \), which means its domain is \( (-\infty, 0) \cup (0, \infty) \). Since the output values (range) can never include \( 0 \), the range is also \( (-\infty, 0) \cup (0, \infty) \). Likewise, the inverse function, \( f^{-1}(x) = \frac{3}{x} \), has the same domain and range as \( f(x) \), resulting in: - Domain of \( f^{-1} \): \( (-\infty, 0) \cup (0, \infty) \) - Range of \( f^{-1} \): \( (-\infty, 0) \cup (0, \infty) \) Now, for the graphing part: if you were to graph \( f(x) \), \( f^{-1}(x) \), and the line \( y = x \): - You would plot the hyperbola of \( f(x) \) in the first and third quadrants, approaching the axes but never touching them. - The same applies to \( f^{-1}(x) \) as it is the same curve. - The line \( y = x \) will create a wonderful symmetry, illustrating how these two functions reflect across this line, highlighting their one-to-one nature! Happy graphing!