AI Tutor
Try dividing the number 11 by a million, a billion, or even a trillion. You will get tiny fractions that creep closer and closer to zero, but can you ever make the fraction actually equal zero?
No matter what nonzero input you test, zero is completely unreachable. This off-limits output leads directly to finding the range, which is the set of all possible output values (yy-values) a function can produce.

The Reciprocal Parent Function

Let's inspect the foundational rational function, f(x)=1xf(x) = \frac{1}{x}.
As xx becomes extremely large in either the positive or negative direction, the fraction 1x\frac{1}{x} shrinks toward zero without ever touching it, forming a horizontal asymptote at the line y=0y = 0.
Because the graph covers every single real height except zero, we record this output set using interval notation as (−∞,0)∪(0,∞)(-\infty, 0) \cup (0, \infty).
The union symbol (∪\cup) connects the separate intervals, telling us that an output can belong to the negative branch or the positive branch, but never zero.
What happens if we slide this entire reciprocal curve to a new position on the coordinate plane?

Shifted Reciprocal Functions

Consider the transformed function g(x)=3x−4+2g(x) = \frac{3}{x - 4} + 2. The constant term +2+2 shifts every output value vertically upward by 22 units.
Since the rational piece 3x−4\frac{3}{x - 4} can never equal 00, the overall value g(x)g(x) can never equal 0+2=20 + 2 = 2.