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Chapter 10: Problem 43

Graph each of the exponential functions. $$ f(x)=2^{|x|} $$

Short Answer

Expert verified
The graph is symmetrical with respect to the y-axis and increases exponentially as x moves away from zero.

Step by step solution

01

Understand the function

The given function is \( f(x) = 2^{|x|} \), which is an exponential function with a base of 2. The exponent is the absolute value of \( x \), which means it is always non-negative.
02

Determine the behavior of the function

Since the exponent is \(|x|\), the function will be symmetrical around the y-axis. For positive \(x\), \(f(x) = 2^{x}\), and for negative \(x\), \(f(x) = 2^{-x} = \frac{1}{2^x}\).
03

Plot key points

Let's plot a few key points to understand the graph: - At \(x = 0\), \(f(x) = 2^{|0|} = 1\).- At \(x = 1\), \(f(x) = 2^1 = 2\) and \(x = -1\), \(f(x) = 2^{|-1|} = 2\).- At \(x = 2\), \(f(x) = 4\) and \(x = -2\), \(f(x) = 4\).This symmetry shows that the function grows rapidly for both positive and negative values of \(x\) as \(x\) moves away from 0.
04

Sketch the graph

Based on the key points and the symmetry of the function around the y-axis, sketch the curve such that:- It passes through \((0, 1)\).- As \(x\) moves away from 0, both positive and negative, the function increases exponentially.- Mirror the right side of the graph across the y-axis to show the symmetry due to the absolute value in the exponent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Value
Absolute value is a mathematical concept that refers to the distance of a number from zero on the number line. Its result is always non-negative.
For any real number \(x\), the absolute value is denoted as \(|x|\). For example:
  • If \(x = 3\), then \(|x| = 3\).

  • If \(x = -3\), then \(|x| = 3\).
This characteristic of always being non-negative is crucial when it's a part of exponential functions.
With the function \(f(x) = 2^{|x|}\), \(|x|\) ensures that whether \(x\) is positive or negative, the exponent is always non-negative.
As a result, the graph will behave similarly for both positive and negative values of \(x\).
Graphing Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent.
The function \(f(x) = 2^{x}\) is a typical example, where 2 is the base. In these functions:
  • When the exponent is positive, the function grows rapidly, leading to a steep curve upwards.

  • When the exponent is negative, the function decreases more gradually towards zero, never reaching it.
To graph \(f(x) = 2^{|x|}\):
  • For \(x > 0\), plot \(f(x) = 2^x\).

  • For \(x < 0\), because of \(|x|\), plot \(f(x) = 2^{-x} = \frac{1}{2^x}\).
The graph starts at the point (0,1) and moves sharply upwards, symmetrically, as \(|x|\) increases.
Symmetry in Graphs
Symmetry in graphs helps reduce the complexity of graphing functions, especially with exponential functions having absolute values.
Symmetry around the y-axis occurs when replacing \(x\) with \(-x\) gives the same function value.
For instance, in \(f(x) = 2^{|x|}\), we observe that:
  • For \(x = 1\), \(f(x) = 2^1 = 2\).

  • For \(x = -1\), \(f(x) = 2^1 = 2\).
This symmetry results because for any \(x\), \(f(-x) = f(x)\). Thus, the graph is mirrored about the y-axis.
This means if you know the graph to the right of the y-axis, you can replicate it to the left.
Therefore, symmetry simplifies understanding and plotting exponential functions with absolute values.

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