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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 33
Chapter 5, Problem 33

Graph each function. ƒ(x) = (1/10)-x

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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: \(a^{-b} = \frac{1}{a^b}\).
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.
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.
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.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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Exponential Functions

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.
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Graphs of Exponential Functions
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