Analyzing Compositions of Functions — Quiz

Question 8 • Mathematics III

If f of x is equal to 1 over x and g of x is equal to x plus 3 , 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 9 to 1 in increments of 1, and the scale on the y axis range from negative 5 to 5 in increments of 1. The curve originates from (negative 7 point 5 comma 0) and crosses the y axis at nonsignificant numbers. The curve lies in the 3rd and 4th quadrants.

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 8 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 2 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 4 comma negative 1).

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 2 to 8 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 (1 comma 4). The other curve opens down and to the left in quadrants 2 and 3 and passes through the point (negative 1 comma 2). The curve crosses the x axis at nonsignificant numbers.

Explanation

For a composition, (f ∘ g)(x) means f(g(x)). Since g(x) = x + 3 and f(x) = 1/x, this gives (f ∘ g)(x) = 1/(x + 3). The graph has a vertical asymptote at x = −3; for example, it passes through (−2, 1) and (−4, −1). Those features match choice C, the graph of 1/(x + 3).

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