Question

the graph of $f(x) = \frac{-1}{x}$ is given below. write the equation of a function $g(x)$ which would shift the graph of $f(x)$ left 6 units and down 5 units.

Explanation:

Step1: Recall horizontal shift rule

To shift a function \( f(x) \) left \( h \) units, we replace \( x \) with \( x + h \) in the function. Here, \( h = 6 \), so for \( f(x)=\frac{-1}{x} \), after left shift, it becomes \( f(x + 6)=\frac{-1}{x + 6} \).

Step2: Recall vertical shift rule

To shift a function down \( k \) units, we subtract \( k \) from the function. Here, \( k = 5 \), so we subtract 5 from \( \frac{-1}{x + 6} \). Thus, \( g(x)=\frac{-1}{x + 6}-5 \).

Answer:

\( g(x)=\frac{-1}{x + 6}-5 \)