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

Verified step by step guidance

1. Identify the given function: \(f(x) = \left(\frac{1}{3}\right)^{-x+1}\). Recognize that this is an exponential function with base \(\frac{1}{3}\) and an exponent of \(-x + 1\).
2. Rewrite the exponent to better understand the function's behavior: \(-x + 1\) can be written as \$1 - x$. So, \(f(x) = \left(\frac{1}{3}\right)^{1 - x}\).
3. Recall that \(\left(\frac{1}{3}\right)^{1 - x} = \left(\frac{1}{3}\right)^1 \cdot \left(\frac{1}{3}\right)^{-x} = \frac{1}{3} \cdot 3^x\), since \(\left(\frac{1}{3}\right)^{-x} = 3^x\). This helps to see the function as \(f(x) = \frac{1}{3} \cdot 3^x\).
4. Determine the domain: Since the function is exponential, the domain is all real numbers, so \(\text{Domain} = (-\infty, \infty)\).
5. Determine the range: Exponential functions with positive bases and real exponents produce positive outputs. Since \$3^x > 0\( for all \)x$, and multiplying by \(\frac{1}{3}\) keeps it positive, the range is \(\text{Range} = (0, \infty)\).

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive real number not equal to 1. In this problem, the function involves a base of 1/3 raised to a linear expression in x. Understanding how the exponent affects the function's growth or decay is essential for graphing.

Domain and Range of Exponential Functions

The domain of an exponential function is all real numbers since any real number can be substituted for x. The range depends on the base and transformations; for positive bases, the range is typically all positive real numbers. Identifying these sets helps describe the function's behavior fully.

Transformations of Functions

Transformations such as shifts, reflections, and stretches affect the graph of a function. Here, the exponent is (-x + 1), which includes a reflection across the y-axis and a horizontal shift. Recognizing these changes helps accurately sketch the graph and understand its shape.