To determine how the graph of [tex]\( g(x) = \frac{1}{x+3} + 6 \)[/tex] compares to the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex], follow these steps:
1. Understand the basic function [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = \frac{1}{x} \)[/tex] is a hyperbola with asymptotes at [tex]\( x=0 \)[/tex] (vertical asymptote) and [tex]\( y=0 \)[/tex] (horizontal asymptote).
2. Identify the transformations in [tex]\( g(x) \)[/tex]:
- The function [tex]\( g(x) = \frac{1}{x+3} + 6 \)[/tex] modifies [tex]\( f(x) \)[/tex] in two distinct ways:
1. Horizontal Shift:
- The expression [tex]\( x+3 \)[/tex] inside the denominator indicates a shift to the left by 3 units. This is because replacing [tex]\( x \)[/tex] with [tex]\( x + 3 \)[/tex] effectively moves the graph in the negative x-direction by 3 units.
2. Vertical Shift:
- The constant term [tex]\( +6 \)[/tex] outside the fraction indicates a vertical shift upwards by 6 units. This is because adding 6 to the function's output increases all y-values by 6.
3. Summarize the transformations:
- Considering the horizontal and vertical shifts together, we know that [tex]\( g(x) = \frac{1}{x+3} + 6 \)[/tex] is the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] shifted left 3 units and up 6 units.
Hence, the graph of [tex]\( g(x) = \frac{1}{x+3} + 6 \)[/tex] is the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] shifted left 3 units and up 6 units.
Thus, the correct answer is:
"The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted left 3 units and up 6 units."
