Graphing Rational Functions — Quiz

Question 5 • Mathematics III

Which graph represents the function f of x is equal to 1 over x minus 1 ?

Answer

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches negative infinity as x approaches negative 1 from the left. Its turning point is near (negative 2 comma negative 1). The branch on the right approaches infinity as x approaches negative 1 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (0 comma 1).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals negative 1 as x approaches negative infinity from the right and approaches negative infinity as x approaches 0 from the left. Its turning point is near (negative 1 comma negative 2). The branch on the right approaches infinity as x approaches 0 from the right and approaches y equals negative 1 as x approaches infinity from the left. Its turning point is near (1 comma 0).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches negative infinity as x approaches 1 from the left. Its turning point is near (0 comma negative 1). The branch on the right approaches infinity as x approaches 1 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (2 comma 1).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals 1 as x approaches negative infinity from the right and approaches negative infinity as x approaches 0 from the left. Its turning point is near (negative 1 comma 0). The branch on the right approaches infinity as x approaches 0 from the right and approaches y equals 1 as x approaches infinity from the left. Its turning point is near (1 comma 2).

Explanation

For f(x) = 1/(x − 1), the denominator is zero at x = 1, so the graph has a vertical asymptote there. As x becomes very large or very negative, the function approaches 0, giving a horizontal asymptote at y = 0. Also, f(0) = −1 and f(2) = 1, matching the graph described in choice C.

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