Graph each function. See Example 2. ƒ(x) = (1/10)^-x

Verified step by step guidance

1. Identify the base of the exponential function: The function given is \( f(x) = \left(\frac{1}{10}\right)^{-x} \). The base of the exponential function is \( \frac{1}{10} \).
2. Rewrite the function using properties of exponents: Since \( a^{-x} = \frac{1}{a^x} \), rewrite the function as \( f(x) = 10^x \).
3. Determine the key characteristics of the function: The function \( f(x) = 10^x \) is an exponential growth function with a base greater than 1, which means it will increase as \( x \) increases.
4. Identify the y-intercept: For exponential functions of the form \( f(x) = a^x \), the y-intercept is at \( (0, 1) \) because \( a^0 = 1 \).
5. Plot the graph: Start by plotting the y-intercept at \( (0, 1) \). Then, choose a few values of \( x \) (e.g., \( x = 1, 2, -1, -2 \)) to calculate corresponding \( f(x) \) values and plot these points. Connect the points with a smooth curve that increases as \( x \) increases.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. These functions exhibit rapid growth or decay depending on the value of 'b'. In the given function f(x) = (1/10)^-x, the base is (1/10), indicating a decay as 'x' increases.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize the behavior of a function. For exponential functions, key points can be calculated by substituting values for 'x' and determining corresponding 'f(x)' values. Understanding how to identify intercepts, asymptotes, and the general shape of the graph is crucial for accurate representation.

Transformations of Functions

Transformations of functions refer to changes made to the graph of a function, such as shifts, stretches, or reflections. In the case of f(x) = (1/10)^-x, the negative exponent indicates a reflection across the y-axis, transforming the graph of the basic exponential function. Recognizing these transformations helps in predicting the graph's behavior without plotting numerous points.