Mathematical Modeling: Part Three — Test

Question 15 • Mathematics III

If f of x is equal to 1 over x and g of x is equal to x plus 4 , which of the following is the graph of open paren g circ f close paren times x ?

Answer

Image with description A coordinate plane with a horizontal x-axis labeled from negative 5 to 5 in increments of 1 unit and a vertical y-axis labeled from negative 7 to 3 in increments of 1 unit, with grid lines every 1 unit. The graph consists of two separate curves. The first curve extends from an upward-pointing arrow near 0 comma 3 along the vertical y-axis, decreases sharply to pass through 1 comma negative 3, and flattens out horizontally to a rightward-pointing arrow near 5 comma negative 3.8. The second curve extends from a downward-pointing arrow near 0 comma negative 7 along the vertical y-axis, increases sharply to pass through negative 1 comma negative 5, and flattens out horizontally to a leftward-pointing arrow near negative 5 comma negative 4.2.

Image with description A coordinate plane with a horizontal x-axis labeled from negative 2 to 8 in increments of 1 unit and a vertical y-axis labeled from negative 5 to 5 in increments of 1 unit, with grid lines every 1 unit. The graph consists of two separate curves. The first curve extends from an upward-pointing arrow near 4 comma 5, decreases sharply to pass through 5 comma 1, and flattens out horizontally to a rightward-pointing arrow near 8 comma 0.5. The second curve extends from a downward-pointing arrow near 4 comma negative 5, increases sharply to pass through 3 comma negative 1, and flattens out horizontally to a leftward-pointing arrow near negative 2 comma negative 0.5.

Image with description A coordinate plane with a horizontal x-axis labeled from negative 5 to 5 in increments of 1 unit and a vertical y-axis labeled from negative 2 to 8 in increments of 1 unit, with grid lines every 1 unit. The graph consists of two separate curves. The first curve extends from an upward-pointing arrow near 0 comma 8 along the vertical y-axis, decreases sharply to pass through 1 comma 5, and flattens out horizontally to a rightward-pointing arrow near 5 comma 4.2. The second curve extends from a downward-pointing arrow near 0 comma negative 2 along the vertical y-axis, increases sharply to pass through negative 1 comma 3, and flattens out horizontally to a leftward-pointing arrow near negative 5 comma 3.8.

Image with description A coordinate plane with a horizontal x-axis labeled from negative 8 to 2 in increments of 1 unit and a vertical y-axis labeled from negative 4 to 5 in increments of 1 unit, with grid lines every 1 unit. The graph consists of two separate curves. The first curve extends from an upward-pointing arrow near negative 4 comma 5, decreases sharply to pass through negative 3 comma 1, and flattens out horizontally to a rightward-pointing arrow near 2 comma 0.5. The second curve extends from a downward-pointing arrow near negative 4 comma negative 5, increases sharply to pass through negative 5 comma negative 1, and flattens out horizontally to a leftward-pointing arrow near negative 8 comma negative 0.5.

Explanation

The composition is (g∘f)(x)=g(f(x)). Since f(x)=1/x and g(x)=x+4, this gives (g∘f)(x)=1/x+4. The graph has a vertical asymptote at x=0, a horizontal asymptote at y=4, and passes through (1,5) and (−1,3). Choice A shows these features.

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