Question

Graph each function. Give the domain and range. See Example 3. ƒ(x) = (1/3)x - 2

Accepted Answer

Identify the base function and its transformation. The given function is $$f(x) = \left(\frac{1}{3}\right)^x - 2$$, which is an exponential function with base $$\frac{1}{3}$$ and a vertical shift downward by 2 units. Recall the general shape and properties of the base function $$g(x) = \left(\frac{1}{3}\right)^x. $$Since $$0 \u003c \frac{1}{3} \u003c 1$$, this is an exponential decay function that approaches zero as $$x$$ approaches infinity and grows without bound as $$x$$ approaches negative infinity. Apply the vertical shift to the base function. The graph of $$f(x) is $$the graph of $$g(x)$$ shifted down by 2 units. This means every $$y-$$value of $$g(x) is $$decreased by 2. Determine the domain of $$f(x). $$Since exponential functions are defined for all real numbers, the domain is all real numbers, expressed as $$(-\infty, \infty). $$Determine the range of $$f(x). $$The base function $$g(x)$$ has range $$(0, \infty)$$, so after shifting down by 2, the range becomes $$(-2, \infty). $$This means the graph approaches the horizontal asymptote $$y = -2$$ but never touches or goes below it.

Graph each function. Give the domain and range. See Example 3. ƒ(x) = (1/3)^x - 2

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Key Concepts

Exponential Functions

Domain and Range of Functions

Graphing Transformations