Problem 76 - Precalculus Mathematics for Calculus

Question

Find the functions \(f \circ g\) and \(g \circ f\) and their domains. $$f(x)=3^{x}, \quad g(x)=x^{2}+1$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 3^{x^2 + 1}\), domain: \(\mathbb{R}\). \((g \circ f)(x) = 9^x + 1\), domain: \(\mathbb{R}\).

Step 1: Understand the Composition

The notation \(f \circ g\) means "\(f\) after \(g\)". This implies we first apply \(g\) to \(x\) and then apply \(f\) to the result of \(g(x)\). Likewise, \(g \circ f\) means "\(g\) after \(f\)"; we first apply \(f\) to \(x\), and then apply \(g\) to the result of \(f(x)\).

Step 2: Define the Function \(f \circ g\)

To find \(f \circ g\), substitute \(g(x) = x^2 + 1\) into \(f\). This gives \((f \circ g)(x) = f(g(x)) = f(x^2 + 1) = 3^{x^2 + 1}\).

Step 3: Find the Domain of \(f \circ g\)

The domain of \(f \circ g\) is determined by the domain of \(g(x)\), since \(f(x) = 3^x\) is defined for all real numbers. \(g(x) = x^2 + 1\) is a polynomial, which is also defined for all real numbers. Thus, the domain of \(f \circ g\) is all real numbers \(\mathbb{R}\).

Step 4: Define the Function \(g \circ f\)

To find \(g \circ f\), substitute \(f(x) = 3^x\) into \(g\). This gives \((g \circ f)(x) = g(f(x)) = g(3^x) = (3^x)^2 + 1\), which simplifies to \(9^x + 1\).

Step 5: Find the Domain of \(g \circ f\)

The domain of \(g \circ f\) is determined by the domain of \(f(x) = 3^x\), which is defined for all real numbers. Therefore, the domain of \(g \circ f\) is also all real numbers \(\mathbb{R}\).

Key Concepts

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. Understanding a function's domain is crucial, as it informs us where a function can be applied without running into mathematical issues, such as division by zero or taking the square root of a negative number.
For example, the function \( f(x) = \frac{1}{x} \) has a domain of all real numbers except zero, because division by zero is undefined.

In the case of polynomial functions, like our \( g(x) = x^2 + 1 \), the domain is all real numbers. This is because polynomials do not have any restrictions like division by zero or negative square roots. They are fully defined across all real numbers.

Exponential functions, such as \( f(x) = 3^x \), also have a domain of all real numbers. There are no input values that make the function undefined. Hence, when we talk about the compositions \( f \circ g \) and \( g \circ f \), both have domains in all real numbers because their component functions are defined everywhere.

Exponential Functions

Exponential functions are a type of mathematical function that feature a constant base raised to a variable exponent. These functions are generally expressed in the form \( f(x) = a^x \), where \( a \) is a positive constant.

One of the most common characteristics of exponential functions is their rapid growth, particularly as \( x \) becomes large. This rapid increase occurs because each increment in \( x \) results in the entire function value being multiplied by the base \( a \).

• **Base Greater Than One**: If \( a > 1 \), the function grows exponentially and increases rapidly.
• **Base Between Zero and One**: If \( 0 < a < 1 \), the function represents exponential decay and decreases as \( x \) increases.

A vital feature of exponential functions is that they are never zero or negative, which implies they are always positive. In the context of our composition, the function \( f(x) = 3^x \) uses base 3, meaning it continuously grows as \( x \) increases.

Polynomials

Polynomials are fundamental components in mathematics, forming expressions consisting of variables, coefficients, and non-negative integer exponents. A general polynomial can be structured as \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \), where the \( a \)s are coefficients.

The highest power of \( x \) in a polynomial determines its degree, which specifies the polynomial's behavior and complexity.

• **Degree 0**: A constant function with no variables.
• **Degree 1**: A linear polynomial, forming a straight line graph.
• **Degree 2**: A quadratic polynomial, which creates parabolic shapes.

In our function \( g(x) = x^2 + 1 \), this polynomial is quadratic with a degree of 2. It implies that the function forms a parabola that opens upwards. Typically, polynomial functions have a domain of all real numbers, making them versatile in their application.

These are frequently used to model various real-world situations because of their flexible shapes and uncomplicated computations.