Problem 24 - Calculus with Concepts in Calculus

Question

Find the domain and rule of \(g \circ f\) and \(f \circ g\). \(f(x)=\frac{1}{x}\) and \(g(x)=x^{2}-3 x-10\)

Step-by-Step Solution

Answer

Domain of \(g \circ f\): \(\mathbb{R} \setminus \{0\}\); Rule: \(\frac{1}{x^2} - \frac{3}{x} - 10\). Domain of \(f \circ g\): \(\mathbb{R} \setminus \{-2, 5\}\); Rule: \(\frac{1}{x^2 - 3x - 10}\).

Step 1: Find the Domain of f(x)

First, identify the domain of the function \(f(x) = \frac{1}{x}\). Since the function involves division by \(x\), \(x\) cannot be zero. Therefore, the domain of \(f(x)\) is all real numbers except zero, or \( \mathbb{R} \setminus \{0\} \).

Step 2: Find the Domain of g(x)

Now, determine the domain of \(g(x) = x^2 - 3x - 10\). This is a polynomial function, and polynomial functions are defined for all real numbers. Therefore, the domain of \(g(x)\) is \(\mathbb{R}\).

Step 3: Compute g(f(x))

To find \(g(f(x))\), substitute \(f(x)\) into \(g\). We have \(g(f(x)) = g\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2 - 3\left(\frac{1}{x}\right) - 10 = \frac{1}{x^2} - \frac{3}{x} - 10\).

Step 4: Domain of g(f(x))

The domain of \(g(f(x))\) is restricted by both the domain of \(f(x)\) and any additional restrictions from substituting \(f(x)\) into \(g\). Since \(\frac{1}{x}\) means \(xeq 0\), the domain of \(g(f(x))\) is also \(\mathbb{R} \setminus \{0\}\).

Step 5: Compute f(g(x))

To find \(f(g(x))\), substitute \(g(x)\) into \(f\). We have \(f(g(x)) = f(x^2 - 3x - 10) = \frac{1}{x^2 - 3x - 10}\).

Step 6: Domain of f(g(x))

The function \(f(g(x))\) includes division by \(x^2 - 3x - 10\). Find when this expression equals zero by solving the quadratic equation: \(x^2 - 3x - 10 = 0\). The solutions are \(x = 5\) and \(x = -2\). The domain of \(f(g(x))\) is thus \(\mathbb{R}\setminus \{-2, 5\}\).

Step 7: Conclusion

The domain of \(g \circ f\) is \(\mathbb{R} \setminus \{0\}\) and the function is given by \(g(f(x)) = \frac{1}{x^2} - \frac{3}{x} - 10\). The domain of \(f \circ g\) is \(\mathbb{R} \setminus \{-2, 5\}\) and the function is given by \(f(g(x)) = \frac{1}{x^2 - 3x - 10}\).

Key Concepts

Domain of a Function

The domain of a function is simply the set of all possible input values that won't cause mathematical issues—like division by zero or taking the square root of a negative number. When working with functions, particularly complex or composite ones, understanding the domain helps in determining which values are applicable.

To find the domain:

• Look for values that might make the denominator zero in rational functions.
• Identify values that might cause negative numbers under square roots if dealing with real numbers.
• Consider which inputs are valid for logarithmic functions (greater than zero).

In the example, we encountered two functions:

• For the function \(f(x) = \frac{1}{x}\), the domain excludes zero since dividing by zero is undefined. Thus, it's all real numbers except zero, written as \( \mathbb{R} \setminus \{0\} \).
• The function \(g(x) = x^2 - 3x - 10\) is a polynomial, and polynomials have domains that include all real numbers, hence \( \mathbb{R} \).

Understanding the domain is crucial especially when these functions are part of a more complex composition of functions.

Function Composition

Function composition is like feeding one function's output into another. The notation \(g \circ f\) means you first apply the function \(f\), then apply \(g\) to its result. It is written as \(g(f(x))\). Similarly, \(f \circ g\) means apply \(g\) first, then \(f\).

The result of a composition depends heavily on the domains of the individual functions, because sometimes the output of the first function isn't suitable as an input for the second. In the original exercise:

• For \(g(f(x))\): Begin by substituting \(f(x)\) into \(g\), resulting in \(g(f(x)) = \frac{1}{x^2} - \frac{3}{x} - 10\). The combination restricts the domain to exclude zero, \( \mathbb{R} \setminus \{0\} \).
• For \(f(g(x))\): Substitute \(g(x)\) into \(f\), resulting in \(f(g(x)) = \frac{1}{x^2 - 3x - 10}\). Here, solve \(x^2 - 3x - 10 = 0\) to find values to exclude \((-2 \text{ and } 5)\), making the domain \( \mathbb{R} \setminus \{-2, 5\} \).

This process showcases the importance of carefully understanding each function's behaviour before and after composition.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of variables. An example, \(g(x) = x^2 - 3x - 10\), is a quadratic polynomial, since its highest exponent is 2.

Key characteristics include:

• Polynomials are defined for all real numbers, meaning their domain is often \( \mathbb{R} \).
• The graph is a smooth, continuous curve with no breaks or holes.
• They can be factored or simplified to find roots, or zeros, which are values of \(x\) where the function equals zero.

For our polynomial \(x^2 - 3x - 10\), factoring it gives \((x - 5)(x + 2)\), revealing its zeros at \(x = 5\) and \(x = -2\). Knowing these roots is essential for identifying domain restrictions when this function is part of a composition with rational functions.

Ultimately, polynomial functions are fundamental building blocks in mathematics, helping us understand rates of change, motion, and even predict outcomes in various scientific fields.