Question

consider the following function. \\( f(x) = -\frac{3}{x} \\) \\( \text{(a)} \\) find the inverse function of \\( f \\). \\( f^{-1}(x) = \\) \\( \text{(b)} \\) graph both \\( f \\) and \\( f^{-1} \\) on the same set of coordinate axes.

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start with the function \( f(x)=-\frac{3}{x} \). Replace \( f(x) \) with \( y \), so we have \( y = -\frac{3}{x} \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = -\frac{3}{y} \).

Step3: Solve for \( y \)

Now we solve the equation \( x = -\frac{3}{y} \) for \( y \). First, multiply both sides by \( y \) to get \( xy=- 3\). Then, divide both sides by \( x \) (assuming \( x
eq0 \)) to obtain \( y = -\frac{3}{x} \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

After solving for \( y \), we replace \( y \) with \( f^{-1}(x) \). So, \( f^{-1}(x)=-\frac{3}{x} \).

Answer:

\( f^{-1}(x)=-\dfrac{3}{x} \)