(a) Write an equation for a rational function \( f \) whose graph is same as the graph of \( y=\frac{1}{x} \) shifted to the right 4 units and down 3 units. (b) Write the domain and range of the function in interval notation. Part: \( \mathbf{0} / \mathbf{2} \) Part 1 of 2 (a) \( f(x) \)

To shift the graph of \( y = \frac{1}{x} \) to the right 4 units, you replace \( x \) with \( x - 4 \), giving \( \frac{1}{x - 4} \). Then, to shift it down 3 units, you subtract 3 from the entire function. The equation for the function \( f \) becomes: \[ f(x) = \frac{1}{x - 4} - 3. \] Now you have a rational function that captures the desired transformations! For the domain, since \( f(x) = \frac{1}{x - 4} - 3 \) is undefined when \( x - 4 = 0 \) (or \( x = 4 \)), the domain is all real numbers except for 4. Thus, the domain is: \[ (-\infty, 4) \cup (4, \infty). \] For the range, the function approaches \( -3 \) but never actually reaches it (as the asymptote), and can take on all other real values. Therefore, the range is: \[ (-\infty, -3) \cup (-3, \infty). \]