Problem 53 - Algebra for College Students

Question

Graph each of the functions. $$ f(x)=\frac{e^{x}+e^{-x}}{2} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = \cosh(x) \) is symmetric about the y-axis and resembles a U-shape.

Step 1: Identify the Type of Function

The function given is \( f(x) = \frac{e^x + e^{-x}}{2} \). This is known as the hyperbolic cosine function, often written as \( f(x) = \cosh(x) \). The hyperbolic cosine function is similar to the cosine function but applies to hyperbolic trajectories.

Step 2: Calculate Key Values

Calculate and note some key values for \( x \) to plot the function: - When \( x = 0 \), \( f(0) = \frac{e^0 + e^0}{2} = 1 \).- When \( x = 1 \), \( f(1) = \frac{e^1 + e^{-1}}{2} \approx 1.543 \).- When \( x = -1 \), \( f(-1) = \frac{e^{-1} + e^{1}}{2} \approx 1.543 \).- As \( x \to \pm \infty \), \( f(x) \to \infty \).

Step 3: Sketch the Graph

Using the key values calculated, start by plotting \( f(0) = 1 \), \( f(1) \approx 1.543 \), and \( f(-1) \approx 1.543 \). The function is symmetric about the y-axis, resembling the shape of a parabola. As \( x \to \pm \infty \), the function rises sharply to infinity, without crossing the x-axis. Draw a smooth curve through these points respecting its even symmetry.

Key Concepts

Graphing Functions

Graphing functions can be a fun exercise when you understand what you're working with. Let’s take a closer look at how to graph the hyperbolic cosine function, \( f(x) = \frac{e^{x} + e^{-x}}{2} \). With any function, the first step is to identify and calculate some key values. This will give you reliable points to mark on your graph.

• Start with the center, often when \( x = 0 \). Here, \( f(0) = 1 \), providing a definite starting point.
• Then calculate values for \( x = 1 \) and \( x = -1 \). Both yield approximately \( f(x) = 1.543 \).

Once you plot these points, observe how the function behaves. As \( x \) grows positively or negatively, notice that the function heads steeply towards infinity. By joining these points smoothly, you can visualize the hyperbolic cosine curve. This approach also highlights the function's unique traits.

Hyperbolic Functions

Hyperbolic functions are fascinating and quite similar to their trigonometric counterparts. However, they deal with hyperbolic angles instead of circular ones. These functions help model situations related to hyperbolas, sort of like how sine and cosine work with circles.

The hyperbolic cosine function \( \cosh(x) = \frac{e^{x} + e^{-x}}{2} \) is one such function. It resembles a curve known as a catenary: the shape assumed by a hanging chain or cable when supported at its ends. Some cool facts about hyperbolic functions include:

• They are closely linked to exponential functions, as seen with \( \cosh(x) \).
• Unlike trigonometric functions, there's no natural periodicity.
• They often appear in real-world models, such as engineering structures and physics.

These features make hyperbolic functions important tools for scientists and engineers.

Symmetry in Functions

Symmetry in functions helps us understand and simplify graphing. Some functions exhibit even symmetry, meaning they mirror equally across the y-axis. The hyperbolic cosine function is one such even function, so \( f(x) = f(-x) \). This indicates its graph reflects identically on both sides of the vertical axis.

The simplicity of symmetry has several benefits:

• Saves time by reducing the need to calculate values for negative and positive \( x \) values separately.
• Makes sketching graphs easier, as you need only know one side well to reflect it properly.
• Shows intrinsic properties about the nature of the function, like continuity and smoothness.

Understanding these concepts makes complex functions more approachable and graphing them straightforward.