Question

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

Accepted Answer

Identify the function given: $$f(x) = \left(\frac{1}{3}\$$right)^{x+2}$$. This is an exponential function with base $$\frac{1}{3}$$, which is between 0 and 1, indicating exponential decay. Determine the domain of the function. Since exponential functions are defined for all real numbers, the domain is all real numbers, expressed as $$(-\infty, \infty)$$. Analyze the range of the function. Because $$\left(\frac{1}{3}\$$right)^{x+2} is $$always positive (a positive base raised to any real power), the function's values are always greater than 0. Therefore, the range is $$(0, \infty). To $$graph the function, start by plotting key points. For example, calculate $$f(x) at$$ $$x = -2 to $$find the y-intercept: $$f(-2) = \left(\frac{1}{3}\$$right)^0$$ = 1. $$Then choose other values like $$x = 0$$ and $$x = -4 to $$see how the function behaves. Sketch the curve showing exponential decay: as $$x$$ increases, $$f(x)$$ approaches 0 but never touches the x-axis (horizontal asymptote at $$y=0$$), and as $$x$$ decreases, $$f(x)$$ grows larger. Label the domain and range on your graph accordingly.

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

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Key Concepts

Exponential Functions

Domain and Range of Functions

Graphing Transformations