Problem 43 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(x\), \(f(x) = 2^{-x} = \frac{1}{2^x}\).

Step 3: 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.

Step 4: 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.

Key Concepts

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.