Analyzing Compositions of Functions — Quiz

Question 9 • Mathematics III

If f of x is equal to 1 over x and g of x is equal to x minus 2 , which of the following is the graph of open paren f circ g close paren times x ?

Answer

Image with description On a coordinate plane, the scale on the x axis range from negative 7 to 2 in increments of 1, and the scale on the y axis range from negative 5 to 5 in increments of 1. The graph is made up of two different curves that do not meet. One of the curves opens up and to the right in quadrants 1 and 2 and passes through the point (negative 1 comma 1). The curve crosses the x axis at nonsignificant numbers. The other curve opens down and to the left in quadrant 3 and passes through the point (negative 3 comma negative 1).

Image with description On a coordinate plane, the x and y axes list from negative 5 to 5 in increments of 1. The graph is made up of two different curves that do not meet. One of the curves opens up and to the right in quadrant 1 and passes through the point (1 comma 1). The other curve opens down and to the left in quadrant 3 and passes through the point (negative 1 comma negative 1).

Image with description On a coordinate plane, the scale on the x axis range from negative 3 to 7 in increments of 1, and the scale on the y axis range from negative 5 to 5 in increments of 1. The graph is made up of two different curves that do not meet. One of the curves opens up to the right in quadrant 1 and passes through the point (3 comma 1). The other curve opens down and to the left in quadrants 3 and 4 and passes through the point (1 comma negative 1). The curve crosses the y axis at nonsignificant numbers.

Image with description On a coordinate plane, the scale on the x axis range from negative 5 to 5 in increments of 1, and the scale on the y axis range from negative 7 to 3 in increments of 1. The graph is made up of two different curves that do not meet. One of the curves opens up to the right in quadrants 1 and 5 and passes through the point (1 comma negative 3). The curve crosses the x axis at nonsignificant numbers. The other curve opens down and to the left in quadrant 3 and passes through the point (negative 1 comma negative 3).

Explanation

The composition is (f ∘ g)(x) = f(g(x)) = 1/(x − 2). It has a vertical asymptote at x = 2 and a horizontal asymptote at y = 0. For example, it passes through (3, 1) and (1, −1), matching choice D; its negative branch lies below the x-axis for x < 2.

AI-written and checked against the verified answer.

More from this Course

Quiz

Quiz

Quiz