Question

100 Points

Which statement about f(x) and its translation, g(x), is true? f(x)=(2/3)^x g(x)=(2/3)^x -3

A. The range g(x) of is, y and the range of f(x) is y .

B. The range g(x) of is, y > 3 and the range of f(x) is y .

C. The asymptote of g(x) is the asymptote of f(x) shifted three units down.

D. The asymptote of g(x) is the asymptote of f(x) shifted three units up.

Answer 1

The correct answer is:

C. The asymptote of g(x) is the asymptote of f(x) shifted three units down.

The function f(x) represents an exponential decay function with a base of 2/3. It does not have an asymptote. However, g(x) is obtained by shifting f(x) vertically downward by 3 units, resulting in a translation of the graph. The asymptote of g(x) would have the same slope as the asymptote of f(x), but it would be shifted downward by 3 units.

Answer 2

C. The asymptote of g(x) is the asymptote of f(x) shifted three units down.

Explanation:

The range of f(x) is y > 0 because
`\left((2)/(3)\right)^x` is always positive.

The range of g(x) is y > -3 because f(x) is always positive and when we subtract 3 from it, it will always be greater than -3.

The asymptote of f(x) is y = 0.

The asymptote of g(x) is y = -3.

Therefore, the correct answer is:

C. The asymptote of g(x) is the asymptote of f(x) shifted three units down.