Rational Functions — Test

Question 22 • Precalculus

Which graph represents the function f of x is equal to the fraction with numerator 2 x and denominator x squared minus 1 ?

Answer

Image with description A coordinate plane with a graph showing three branches. The branch on the left approaches y equals 2 as x approaches negative infinity from the right and approaches infinity as x approaches negative 1 from the left. The branch in the middle approaches negative infinity as x approaches negative 1 from the right and approaches negative infinity as x approaches 1 from the left. The branch on the right approaches infinity as x approaches 1 from the right and approaches y equals 2 as x approaches infinity from the left.

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. The branch on the right approaches negative infinity as x approaches 1 from the right and approaches y equals 1 as x approaches infinity from the left.

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals 2 as x approaches negative infinity from the right and approaches negative infinity as x approaches 1 from the left. The branch on the right approaches infinity as x approaches 1 from the right and approaches y equals 2 as x approaches infinity from the left.

Image with description A coordinate plane with a graph showing three 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. The branch in the middle approaches infinity as x approaches negative 1 from the right and approaches negative infinity as x approaches 1 from the left. It has a turning point at the origin. 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.

Explanation

The denominator x² − 1 is zero at x = −1 and x = 1, so the graph has vertical asymptotes there. Since the denominator’s degree is greater than the numerator’s, the horizontal asymptote is y = 0. Between −1 and 1, the graph goes from positive infinity to negative infinity and crosses the origin, where f(0) = 0. These features match choice D.

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