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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 37
Chapter 5, Problem 37

Graph each function. See Example 2. ƒ(x) = 2|x|

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Understand the function given: \(f(x) = 2^{|x|}\). This means the output is 2 raised to the power of the absolute value of \(x\).
Recall that the absolute value function \(|x|\) makes all \(x\) values non-negative, so for negative \(x\), \(|x| = -x\), and for non-negative \(x\), \(|x| = x\).
Rewrite the function as a piecewise function to understand its behavior on both sides of the y-axis: \(f(x) = \begin{cases} 2^x & \text{if } x \geq 0 \\ 2^{-x} & \text{if } x < 0 \end{cases}\)
Graph the right side (\(x \geq 0\)) using the exponential function \$2^x\(, which increases as \)x$ increases.
Graph the left side (\(x < 0\)) using \(2^{-x}\), which is the reflection of \$2^x$ across the y-axis, ensuring the graph is symmetric about the y-axis.

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Key Concepts

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

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x, making all inputs positive or zero. It creates a V-shaped graph symmetric about the y-axis, affecting how the function behaves for negative and positive inputs.
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Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is positive and not equal to 1. They exhibit rapid growth or decay, and their graphs pass through (0,1), reflecting the property that any number to the zero power equals one.
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Graphing Composite Functions

Graphing composite functions like f(x) = 2^{|x|} involves understanding how the inner function (absolute value) transforms the input before applying the outer function (exponential). This results in a graph symmetric about the y-axis, combining properties of both functions.
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