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Answer:
f(x) = (3x -12)/(x² -7x +12)
Step-by-step explanation:
The equation for a rational function can be written from the clues offered by the horizontal and vertical asymptotes, the y-intercept, and any "holes" in the function graph. Here, those clues are ...
- horizontal asymptote: y = 0
- vertical asymptote: x = 3
- hole: (x, y) = (4, 3)
- y-intercept: (0, -1)
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The horizontal asymptote at y=0 tells you the denominator is of higher degree than the numerator. The vertical asymptote at x=3 tells you (x-3) is a factor of the denominator. The hole at x=4 tells you there is a factor of (x-4) in both numerator and denominator.
At this point, the function is tentatively ...
Evaluating this at the y-intercept, x=0, gives ...
f(0) = -1 = a/(0 -3) = -a/3
a = 3 . . . . . multiply by -3
Then our rational function, consistent with all of the features of the graph, is ...