Question

Define two other transcendental functions. The hyperbolic sine function, designated by \(\sinh x,\) is defined as $$ \sinh x=\frac{1}{2}\left(e^{x}-e^{-x}\right) $$ (a) Show that \(f(x)=\sinh x\) is an odd function. (b) Graph \(f(x)=\sinh x\)

Short answer

\(\sinh x\) is an odd function, and its graph is sigmoid-shaped, symmetric around the origin.

Step-by-step solution

- Define an odd function

A function is said to be odd if it satisfies the condition: \[ f(-x) = -f(x) \]for all values of x in the function's domain.

- Substitute \(-x\) into \( \sinh x \)

Calculate \( \sinh(-x) \) using the definition of the hyperbolic sine function:\[\sinh(-x) = \frac{1}{2}(e^{-x} - e^{x})\]

- Simplify the expression for \ \sinh(-x) \

Factor out a \ -1 \ from the expression inside the parentheses:\[\sinh(-x) = \frac{1}{2}(-1)(e^{x} - e^{-x}) = -\frac{1}{2}(e^{x} - e^{-x}) = -\sinh x\]

- Conclude odd function property

Since \( \sinh(-x) = -\sinh(x) \), \( \sinh x \) is an odd function.

- Plot the graph \( \( \sinh x \)\)

To graph \(\sinh x\), plot a few key points. Note that: When \(x = 0\), \(\sinh 0 = 0\).When \(x = 1\), \(\sinh 1 \approx 1.175\).When \(x = -1\), \(\sinh -1 \approx -1.175\).

- Draw the characteristic shape of the hyperbolic sine function

The graph of \(\sinh x\) is sigmoid-shaped and crosses the origin at (0,0), being symmetric around the origin due to its odd function properties. It increases exponentially for positive x and decreases exponentially for negative x.

Key concepts

Odd Function

An odd function is a function that satisfies the equation \[ f(-x) = -f(x) \] for all x in its domain. This property means that the graph of the function is symmetric with respect to the origin. In simpler terms, flipping the graph over both the x-axis and y-axis will result in the same graph we started with. We showed that the hyperbolic sine function \[ \text{sinh}(x) \] is odd by verifying \[ \text{sinh}(-x) = -\text{sinh}(x) \]. So, if you calculate \[ \text{sinh}(-2) \], you should get \[ -\text{sinh}(2) \].

Transcendental Functions

Transcendental functions are non-algebraic functions that are not solutions to polynomial equations with integer coefficients. Examples include exponential functions, logarithmic functions, and trigonometric functions. They often arise in complex situations involving growth, decay, and oscillations. The hyperbolic sine function \[ \text{sinh}(x) \] is a transcendental function because it incorporates the exponential function \[ e^x \], which itself is transcendental. These functions cannot be expressed in terms of a finite combination of algebraic operations.

Graphing Hyperbolic Functions

Graphing hyperbolic functions like \[ \text{sinh}(x) \] involves understanding their shapes and key points. To graph \[ \text{sinh}(x) \], start by noting a few points:

• When \[ x = 0, \text{sinh}(0) = 0 \].
• When \[ x = 1, \text{sinh}(1) \] is approximately 1.175.
• When \[ x = -1, \text{sinh}(-1) \] is approximately -1.175.

You can see that the graph crosses the origin and follows a symmetric sigmoid shape, growing exponentially as x increases and decreasing exponentially as x becomes more negative. This symmetry around the origin comes from \[ \text{sinh}(x) \] being an odd function.

Exponential Growth and Decay

Exponential growth and decay describe processes that increase or decrease at rates proportional to their current value. In the graph of \[ \text{sinh}(x) \], you'll notice that as x becomes large, the function grows rapidly because it includes the term \[ e^x \]. Conversely, as x becomes very negative, \[ \text{sinh}(x) \] also decreases rapidly due to the term \[ e^{-x} \]. This behavior captures both exponential growth for positive x and exponential decay for negative x. Such properties make hyperbolic functions useful in various scientific fields, from physics to engineering.