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 2 units and up 4 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, we shift left 2 units, so \( h = 2 \). So for \( f(x)=\frac{1}{x} \), after horizontal shift, it becomes \( f(x + 2)=\frac{1}{x + 2} \).

Step2: Recall vertical shift rule

To shift a function \( f(x) \) up \( k \) units, we add \( k \) to the function. Here, we shift up 4 units, so \( k = 4 \). So we add 4 to the horizontally shifted function.

Combining both shifts, the function \( g(x) \) is \( g(x)=\frac{1}{x + 2}+4 \).

Answer:

\( g(x)=\frac{1}{x + 2}+4 \)