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

Verified step by step guidance

1. Identify the base function and transformations. The base function here is an exponential function of the form \(f(x) = a^{x}\), specifically \(f(x) = \left(\frac{1}{3}\right)^x\). The given function is \(f(x) = -\left(\frac{1}{3}\right)^{x+2} - 1\), which involves transformations of the base function.
2. Analyze the horizontal shift. The expression \(x + 2\) inside the exponent indicates a horizontal shift to the left by 2 units. This means the graph of \(\left(\frac{1}{3}\right)^x\) is shifted left 2 units.
3. Consider the reflection and vertical shift. The negative sign in front of the exponential, \(-\left(\frac{1}{3}\right)^{x+2}\), reflects the graph across the x-axis. Then, subtracting 1, as in \(-\left(\frac{1}{3}\right)^{x+2} - 1\), shifts the graph down by 1 unit.
4. Determine the domain. Since exponential functions are defined for all real numbers, the domain of \(f(x)\) is all real numbers, expressed as \((-\infty, \infty)\).
5. Find the range by considering the transformations. The base function \(\left(\frac{1}{3}\right)^x\) has a range of \((0, \infty)\). After reflection and vertical shift, the range becomes \((-\infty, -1)\) because the graph is reflected over the x-axis and shifted down by 1.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the variable is in the exponent. Understanding how the base affects growth or decay is essential. In this problem, the function involves (1/3) raised to a power, indicating exponential decay since the base is between 0 and 1.

Transformations of Functions

Transformations include shifts, reflections, and stretches/compressions of the graph. Here, the function has a negative sign in front, indicating a reflection over the x-axis, and the exponent includes a horizontal shift by -2. Recognizing these helps in accurately graphing the function.

Domain and Range of Exponential Functions

The domain of exponential functions is all real numbers since any real number can be an exponent. The range depends on transformations; for example, reflections and vertical shifts change the range. Identifying these helps determine the set of possible output values for the function.