Graph each function. ƒ(x) = (1/10)^-x

Verified step by step guidance

1. Recognize that the function is given by \(f(x) = \left(\frac{1}{10}\right)^{-x}\). The negative exponent means we can rewrite the function using the property of exponents: \(a^{-b} = \frac{1}{a^b}\).
2. Rewrite the function as \(f(x) = \left(\frac{1}{10}\right)^{-x} = 10^x\). This simplifies the function to an exponential growth function with base 10.
3. Identify key points to plot the graph by substituting values for \(x\). For example, calculate \(f(0)\), \(f(1)\), \(f(-1)\), and \(f(2)\) to get points on the graph.
4. Plot the points on the coordinate plane using the values found, and draw a smooth curve through these points. Since \(f(x) = 10^x\) is an exponential growth function, the graph will increase rapidly as \(x\) increases.
5. Note the horizontal asymptote of the graph. For \(f(x) = 10^x\), the graph approaches the \(x\)-axis (or \(y=0\)) but never touches it as \(x\) approaches negative infinity.

Key Concepts

Exponents and Negative Exponents

Exponents indicate how many times a base is multiplied by itself. A negative exponent means taking the reciprocal of the base raised to the positive exponent, for example, a^(-x) = 1/(a^x). Understanding this helps simplify and interpret expressions like (1/10)^-x.

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. These functions model growth or decay and have distinctive graphs that either increase or decrease exponentially depending on the base and exponent.

Graphing Exponential Functions

Graphing involves plotting points by substituting values of x into the function and understanding the shape of the curve. Key features include the y-intercept at (0,1), asymptotes, and whether the function is increasing or decreasing based on the base and exponent sign.