Problem 67 - Precalculus Essentials

Question

Find; a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\frac{2}{x+3}, g(x)=\frac{1}{x}$$

Step-by-Step Solution

Answer

a. The composite function \(f \circ g(x)=\frac{2x}{1+3x}\)\n b. The domain of the composite function is \(-\infty,-\frac{1}{3}) \cup (-\frac{1}{3},+\infty)\)

Step 1: Find the composite function \(f \circ g\)

Firstly, calculate \(f \circ g\) which means applying g(x) into f(x). Replace x in f(x) with the function g(x), this gives \(f(g(x))=f \left( \frac{1}{x} \right)=\frac{2}{\frac{1}{x}+3}\)

Step 2: Simplify the function

Simplify the function \(f(g(x))\). This results in \(f(g(x))=\frac{2}{\frac{1+3x}{x}}\), which simplifies further to \(f(g(x))=\frac{2x}{1+3x}\)

Step 3: Determine the domain of \(f \circ g\)

Secondly, find the domain of \(f \circ g\). A rational function is undefined when the denominator equals zero. Thus, to find the domain of the function, set the denominator \(1+3x\) equal to zero and solve for x. Solving \(1+3x = 0\) yields \(x=-\frac{1}{3}\). Therefore, the domain of the function \(f \circ g\) is all real numbers except \(x=-\frac{1}{3}\). We express this in interval notation as \(-\infty,-\frac{1}{3}) \cup (-\frac{1}{3},+\infty)\)

Key Concepts

Domain of a Function

The domain of a function is a collection of all possible input values (usually x-values) that a function can accept without causing any mathematical errors, like division by zero. When dealing with composite functions, finding the domain can involve a combination of restrictions from these functions.

1. **Identify Restrictions in Individual Functions:**

• For the function \(g(x) = \frac{1}{x}\), we have an issue when \(x = 0\) since division by zero is not defined.
• For \(f(x) = \frac{2}{x+3}\), we face a problem when \(x + 3 = 0\) or when \(x = -3\).

2. **Apply Composition:** When composing \(f \circ g(x)\), the value from \(g(x)\) needs to also satisfy \(f(x)\)'s domain.

3. **Identify Overall Restrictions in Composite Function:** For \(f(g(x)) = \frac{2x}{1+3x}\), you'll encounter division by zero when \(1+3x = 0\). This restriction leads to \(x = -\frac{1}{3}\).

So, the domain of the composite function \(f \circ g\) excludes these x-values that cause division by zero, represented as \(-\infty, -\frac{1}{3}) \cup (-\frac{1}{3}, +\infty)\). Always check each function's restrictions before and after composition!

Rational Functions

Rational functions are expressions that involve the division of two polynomials. A typical form of a rational function is \( \frac{p(x)}{q(x)} \), where both \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\). The denominator is not allowed to be zero, which is a key factor in determining a function’s domain.

**Features of Rational Functions:**

• Asymptotes: These functions often have vertical asymptotes where the function is undefined (denominator equals zero).
• Intercepts: Rational functions can have horizontal and vertical intercepts, providing points where the graph crosses axes.
• Simplification: Rational functions can sometimes be simplified by canceling common factors in the numerator and denominator.

In our problem, the rational function constructed through function composition \( \frac{2x}{1+3x} \) is a typical example. The denominator \(1+3x\) should not be zero, indicating where to find exclusions in the domain.

Understanding these characteristics helps in graphing these functions and predicting behavior near points of discontinuity.

Function Composition

Function composition is a method of combining two functions, whereby the output of one function becomes the input of another. It's written as \((f \circ g)(x)\) which reads "f composed with g of x".

**Steps in Function Composition:**

• Substitute: Replace every instance of x in \(f(x)\) with \(g(x)\).
• Simplify: After substitution, simplify the resulting expression to find the composite function.

In our exercise, we begin with substituting \(g(x) = \frac{1}{x}\) into \(f(x) = \frac{2}{x+3}\) to get \( f(g(x)) = f \left( \frac{1}{x} \right) = \frac{2}{\frac{1}{x} + 3}\).

Simplifying it further gives \(\frac{2x}{1+3x}\), which is a new function formed by the composition of \(f\) and \(g\). Understanding function composition is critical in many areas of mathematics, including solving algebraic equations and modeling real-life scenarios with complex data. It allows you to see how two processes or transformations can be combined to yield a new result, revealing more about variable interactions.