Question

If \(f(x)=\frac{\sqrt{25 x^{2}-4}}{x}\) and \(g(x)=\frac{2}{5} \sec x, 0

Short answer

\((f \circ g)(x)=5 \sin x\)

Step-by-step solution

- Understand given functions

Identify the two functions given. First function: \(f(x)=\frac{\sqrt{25 x^{2}-4}}{x}\). Second function: \(g(x)=\frac{2}{5} \sec x\).

- Define composition of functions

Composition of \(f\) and \(g\) means \((f \circ g)(x) = f(g(x))\). Substitute \(g(x)\) into \(f(x)\).

- Substitute \(g(x)\) into \(f(x)\)

Substitute \(g(x) = \frac{2}{5} \sec x\) into \(f(x)\). We get: \(f\left(\frac{2}{5} \sec x\right) = \frac{\sqrt{25 \left(\frac{2}{5} \sec x\right)^{2} - 4}}{\frac{2}{5} \sec x}\).

- Simplify the expression inside the square root

Simplify: \[ 25 \left(\frac{2}{5} \sec x\right)^{2} = 25 \cdot \frac{4}{25} \sec^{2} x = 4 \sec^{2} x \]. Then the expression inside the square root is: \[ \sqrt{4 \sec^{2} x - 4} = \sqrt{4 (\sec^{2} x - 1)} \].

- Simplify further using trigonometric identity

Use identity \(\sec^{2} x - 1 = \tan^{2} x\). Thus, \(\sqrt{4 (\sec^{2} x - 1)} = \sqrt{4 \tan^{2} x} = 2 \left| \tan x \right|\). Since \(0 < x < \frac{\pi}{2}\), \(\tan x\) is positive, so \(2 \left| \tan x \right| = 2 \tan x\).

- Complete the composition

Now, \(f\left( \frac{2}{5} \sec x \right) = \frac{2 \tan x}{\frac{2}{5} \sec x}\). Simplify the fraction: \[ \frac{2 \tan x}{\frac{2}{5} \sec x} = 5 \tan x \cdot \cos x = 5 \sin x. \] Thus, \((f \circ g)(x)=5 \sin x\).

Key concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially when solving trigonometric equations and simplifying expressions. One such identity used in this exercise is the Pythagorean identity for secant. For any angle \(x\):
- \( \text{sec}^2 x - 1 = \text{tan}^2 x \).
This identity simplifies the radical expression \( \text{sec}^2x - 1 \) into \( \text{tan}^2 x \), which is easier to manage. Identifying and using these identities lets us transform and simplify complex expressions in trigonometric functions, leading to straightforward results. Mastering these identities helps in solving various trigonometric functions and equations effectively.

Function Composition

Function composition is when we apply one function to the results of another, effectively creating a new function. We denote this composition by \((f \, \text{◦} \, g)(x)\), which means \(f\) applied to \(g(x)\).
In the given exercise, the composition of functions \(f\) and \(g\) is represented as \((f \, \text{◦}\, g)(x)\). To achieve this, we substitute \(g(x)\) into the function \(f(x)\). For our exercise:
- \(f(x) = \frac{\text{√}(25 x^2 - 4)}{x}\)
- \(g(x) = \frac{2}{5} \text{sec} x\).
The composition \( (f \text{◦} g)(x)\) then translates to \(f(g(x))\), showing how the two functions interact when combined. This concept lets us build new functions from existing ones, offering a powerful method for solving complex math problems.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra and trigonometry. It involves reducing a complicated expression into its simplest form while maintaining its value. In our exercise, we start with the composed function:
- \(f\big( \frac{2}{5} \text{sec} x \big) = \frac{\text{√}(25 (\frac{2}{5} \text{sec} x)^2 - 4)}{\frac{2}{5} \text{sec} x}\).
To simplify, follow these steps:
1. Simplify the expression inside the square root: \(25 \big( \frac{2}{5} \text{sec} x \big)^2\).
2. Use trigonometric identities to further simplify: \( 4 \text{sec}^2 x - 4\).
3. Substitute \( \text{sec}^2 x - 1 = \text{tan}^2 x\) to further reduce the expression to \( 4 \text{tan}^2 x \).
4. Simplify the square root operation: \(2 |\text{tan}x| = 2 \text{tan} x\).
5. Finally, simplify the overall equation.
Each step reduces the complexity of the expression, making it more manageable and easier to solve.

Square Roots in Algebra

Square roots frequently appear in algebra, and effectively working with them often requires simplification and the use of identities. In this problem, we encounter a square root within the function \(f(x)\).
- \( \text{√} (25 x^2 - 4)\)
First, recognize the importance of properly distributing operations inside and outside the square root. In this case, we simplify the expression inside the square root before taking the square root itself. Use algebraic tricks and identities like Pythagorean identities to restructure expressions into simpler forms like:
- \(\text{√}(4 \text{tan}^2 x) = 2 \text{tan} x\).
Understanding how to manipulate square roots and their related expressions will improve problem-solving skills and the ability to simplify complex algebraic and trigonometric functions.