Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$f(x)=3^{x}+3^{-x}$$

Step-by-Step Solution

Answer

The graph of the function is symmetric about the y-axis, with a minimum at \( x=0 \) and range \([2, \infty)\).

Step 1: Identify the Type of Function

The function given is \( f(x) = 3^x + 3^{-x} \). This is an exponential function because it involves terms with variable exponents. More specifically, it includes both \( 3^x \) and its reciprocal \( 3^{-x} \).

Step 2: Find the Domain

The domain of \( f(x) = 3^x + 3^{-x} \) is all real numbers because exponential functions are defined everywhere on the real number line. Thus, the domain is \( (-\infty, \infty) \).

Step 3: Determine the Range

The smallest value of the expression \( 3^x + 3^{-x} \) occurs at \( x = 0 \). Evaluating at \( x = 0 \), we get \( f(0) = 3^0 + 3^0 = 1 + 1 = 2 \). Therefore, the minimum value is 2. As \( x \to \infty \), \( 3^{-x} \to 0 \) and \( f(x) \approx 3^x \to \infty \). Hence, the range is \([2, \infty)\).

Step 4: Identify Symmetry and Asymptotes

The function \( f(x) = 3^x + 3^{-x} \) is an even function because \( f(-x) = f(x) \). It is symmetric about the y-axis. There are no horizontal or vertical asymptotes as both terms grow without bound as \( x \to \infty \).

Step 5: Calculate Important Points

Calculate \( f(x) \) at important points to assist with graphing: \( f(0) = 2 \), \( f(1) = 3^1 + 3^{-1} = 3 + \frac{1}{3} = \frac{10}{3}\), and \( f(-1) = 3^{-1} + 3^{1} = \frac{1}{3} + 3 = \frac{10}{3} \).

Step 6: Sketch the Graph

Plot the points found in the previous step and note the symmetry along the y-axis. Connect these points smoothly, considering that the graph will rise more steeply as \( x \to \infty \) and \( x \to -\infty \) because of the exponential growth of \( 3^x \) and \( 3^{-x} \). The minimum value occurs at \( x = 0 \) with a value of 2.

Key Concepts

Even Functions

Even functions are fascinating mathematical entities. An even function is defined by the property that for any value of x in its domain, the value at negative x is the same. In other words,

• \( f(-x) = f(x) \).

This definition means an even function is symmetric with respect to the y-axis. A mirror reflection along the y-axis will look identical. For the given function, \( f(x) = 3^x + 3^{-x} \), we can see that it satisfies this condition since

• \( f(-x) = 3^{-x} + 3^x = f(x) \).

This symmetry means if you were to flip the graph over the y-axis, it would match up perfectly with itself. This unique characteristic makes even functions particularly easy to sketch when you are aware of just one side of the graph.

Domain and Range

Understanding the domain and range of a function is crucial for graphing and solving equations. The

• Domain of a function refers to all the possible input values (x-values) that the function can accept without causing any mathematical issues such as dividing by zero or taking the square root of a negative number.

For the function \( f(x) = 3^x + 3^{-x} \), the domain is all real numbers, or

• \( (-\infty, \infty) \), because exponential functions are defined for every real number without any restriction.

The

• Range of a function is the set of all possible output values (y-values) as x varies over its domain.

For this particular function, the range is

• \([2, \infty) \).This means the smallest value taken by the function is 2, which occurs at \( x = 0 \). As x increases indefinitely in either direction, the value of the function approaches infinity.

Symmetry

Symmetry is a critical aspect of understanding and graphing functions. Symmetry about the y-axis means that one side of the graph is the mirror image of the other. For even functions such as \( f(x) = 3^x + 3^{-x} \), this y-axis symmetry is inherent in their structure.

• To determine symmetry, one often substitutes \( -x \) for \( x \) and checks if the resulting function is identical to the original.

For our function,

• \( f(x) = 3^x + 3^{-x} \) is the same as\( f(-x) \), affirming its symmetry about the y-axis.

In practical terms, this symmetry simplifies graphing the function. If you know the function's behavior for positive x, you automatically know it for negative x as well due to this mirror-image symmetry.

Graphing Functions

Graphing functions involves plotting points and understanding the behavior of the function over its domain. To graph the function\( f(x) = 3^x + 3^{-x} \), we use several steps:

• Identify important points, such as where the function reaches its minimum or maximum value.

In this case,

• \( f(0) = 2 \) is a crucial point, as it is the minimum value of the function.
• Evaluate the function at other critical points like \( f(1) \) and \( f(-1) \).

You can calculate these to be

\( \frac{10}{3} \), allowing you to plot them on the graph.

Because the function is even, you need only calculate points for positive x, as these will mirror for negative x due to the y-axis symmetry. Asx approaches infinity in either direction, remember that \( f(x) \) also increases towards infinity. Thus, after plotting the key points, draw a smooth curve through them, accounting for symmetrical properties and the direction in which the function's values are growing.