Problem 43 - Precalculus with Calculus Previews

Question

Show that \(f(x)=\frac{1}{2}\left(3^{x}+3^{-x}\right)\) is an even function. Sketch the graph of \(f\).

Step-by-Step Solution

Answer

The function is even; it is symmetric about the y-axis.

Step 1: Define an Even Function

A function is even if it satisfies the condition \(f(-x) = f(x)\) for all \(x\) in the domain of the function. This means that the graph of the function is symmetric with respect to the y-axis.

Step 2: Substitute \(-x\) into \(f(x)\)

Substitute \(-x\) into the function \(f(x) = \frac{1}{2}(3^x + 3^{-x})\) and evaluate it: \[ f(-x) = \frac{1}{2}(3^{-x} + 3^x) = \frac{1}{2}(3^x + 3^{-x}) = f(x) \].

Step 3: Confirm Even Function

Since \(f(-x) = f(x)\), we established that \(f(x)\) is an even function. Thus, it has symmetry about the y-axis.

Step 4: Sketch the Graph of \(f(x)\)

To sketch the graph of \(f(x) = \frac{1}{2}(3^x + 3^{-x})\), note the following characteristics: - It is an even function, so it is symmetric about the y-axis.- At \(x=0\), \(f(0) = \frac{1}{2}(3^0 + 3^0) = 1\).- As \(x\) approaches \(\infty\), \(f(x)\) grows exponentially, and similarly as \(x \to -\infty\), \(f(x)\) also grows exponentially since \(3^{-x} = \frac{1}{3}^x\) becomes very large.

Key Concepts

function symmetry

A function is considered symmetric when its graph looks the same both to the left and right of a specific line, often the y-axis. When we say a function is symmetric with respect to the y-axis, it's specifically referred to as an **even function**. The mathematical condition that defines an even function is:

• The function must satisfy the equation: \( f(-x) = f(x) \) for all values of \( x \) in its domain.

For example, consider the function given as \( f(x) = \frac{1}{2}(3^x + 3^{-x}) \). When we substitute \(-x\) into this function, we find that:

• \( f(-x) = \frac{1}{2}(3^{-x} + 3^x) = \frac{1}{2}(3^x + 3^{-x}) = f(x) \)

This verification shows the function is even, ensuring its symmetry with respect to the y-axis.

exponential growth

Exponential growth describes a situation where a quantity increases at a rate proportional to its current value. In mathematics, functions like \( 3^x \) are classic examples of exponential growth. As \( x \) increases, \( 3^x \) grows rapidly, exhibiting the characteristic "hockey stick" shape on a graph.

For the function \( f(x) = \frac{1}{2}(3^x + 3^{-x}) \), **both positive and negative exponential terms exist**:

• The term \( 3^x \) grows exponentially as \( x \) becomes large, moving to the right on the x-axis.
• Similarly, \( 3^{-x} = \frac{1}{3^x} \) grows exponentially in importance as \( x \) becomes a large negative number, moving to the left on the x-axis.

This interplay of exponential growth in both directions contributes to the graph's unique shape, first decreasing towards zero at \( x = 0 \) and then increasing towards infinity as \( x \) moves away from zero in either direction.

y-axis symmetry

Y-axis symmetry means that for every point on the graph at position \( (x, y) \), there is a corresponding point at \( (-x, y) \). Essentially, this indicates that the part of the graph to the left of the y-axis is a mirror image of the part to the right.

For the function \( f(x) = \frac{1}{2}(3^x + 3^{-x}) \), it is clear due to its even nature and function properties that it has a y-axis symmetry:

• This symmetry is observed because the expressions \( 3^x \) and \( 3^{-x} \) will yield the same results when \( x \) is replaced with \(-x \).

In practical terms, this means if we were to "fold" the graph along the y-axis, the two sides would perfectly align. This quality is not just a mathematical curiosity, but offers valuable insights into the behavior and graphing of even functions.