Problem 76 - Algebra and Trigonometry

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

Step 1: Define Composition of Functions

To find the composition \(f \circ g\), you will substitute the function \(g(x)\) into the function \(f(x)\). This means you will replace \(x\) in \(f(x)\) with \(g(x)\), so \(f \circ g\) becomes \(f(g(x)) = 3^{g(x)}\).

Step 2: Calculate \(f \circ g\)

Given \(g(x) = x^2 + 1\), substitute \(g(x)\) into \(f(x)\): \[f(g(x)) = f(x^2 + 1) = 3^{x^2 + 1}\] Thus, \(f \circ g(x) = 3^{x^2 + 1}\).

Step 3: Define Domain of \(f \circ g\)

The domain of \(f \circ g\) is determined by the domain of \(g(x)\) since \(f\) can take any real number as an exponent for 3. \(g(x) = x^2 + 1\) is defined for all \(x \in \mathbb{R}\), so the domain of \(f \circ g\) is \(\mathbb{R}\).

Step 4: Define Composition of Functions \(g \circ f\)

To find \(g \circ f\), substitute the function \(f(x)\) into the function \(g(x)\). This means replace \(x\) in \(g(x)\) with \(f(x)\), so \(g \circ f\) becomes \(g(f(x)) = (f(x))^2 + 1\).

Step 5: Calculate \(g \circ f\)

Given \(f(x) = 3^x\), substitute \(f(x)\) into \(g(x)\): \[g(f(x)) = g(3^x) = (3^x)^2 + 1 = 3^{2x} + 1\] Thus, \(g \circ f(x) = 3^{2x} + 1\).

Step 6: Define Domain of \(g \circ f\)

The domain of \(g \circ f\) is determined by the domain of \(f(x)\) since \(g\) can take any real number as input. \(f(x) = 3^x\) is defined for all \(x \in \mathbb{R}\), so the domain of \(g \circ f\) is \(\mathbb{R}\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial because it tells you the set of possible input values, usually called "x", which will produce valid outputs. In the context of function composition, knowing the domain ensures that we are only using inputs that work within both functions.
When dealing with composed functions like \(f \circ g\) or \(g \circ f\), the domain is primarily influenced by the inner function's domain. For example, in \(f \circ g\), we check for \(g(x)\)'s domain, as this function feeds directly into \(f(x)\).

• For \(f(x) = 3^x\), the domain is all real numbers \(x \in \mathbb{R}\), as any real number can be an exponent in an exponential function.
• For \(g(x) = x^2 + 1\), the domain is also all real numbers \(x \in \mathbb{R}\), since squaring and adding a constant will work for all real numbers.

Hence, the domains for both \(f \circ g\) and \(g \circ f\) are all real numbers \(\mathbb{R}\). This is because the range of the inner function is compatible with the domain requirements of the subsequent function.

Exponential Function

An exponential function is one where the variable is in the exponent, such as \(f(x) = a^x\), where \(a\) is a positive constant. These functions are known for their rapid growth as the variable \(x\) increases or decreases.
For the function \(f(x) = 3^x\), it is an exponential function with the base of 3. Exponential functions like this start increasing slowly and then speed up, growing significantly as \(x\) becomes larger.
This growth pattern is so rapid because the rate of increase is proportional to its current value—a process known as exponential growth.

• The domain of an exponential function, \(3^x\), is all real numbers, \(x \in \mathbb{R}\).
• The range is always positive numbers, \(y > 0\), since a positive number raised to any power is always positive.

Exponential functions are fundamental in various fields, including biology, physics, and finance, because they model real-world phenomena where things grow or decay rapidly.

Polynomial Function

Polynomial functions are expressions involving terms made up of variables raised to whole number exponents. An example is the quadratic polynomial \(g(x) = x^2 + 1\). In this function, \(x^2\) represents the variable \(x\) squared, plus a constant term.
These functions have the form \(a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\). Each "a" represents a coefficient, and "n" is a non-negative integer determining its degree.
For \(g(x) = x^2 + 1\):

• It is a quadratic polynomial, which is a specific type of polynomial where the highest power is 2.
• The graph of a quadratic is a parabola, which opens upwards since the coefficient of \(x^2\) is positive.
• The domain is all real numbers \(x \in \mathbb{R}\), because any real number can be squared and then added to another real number.

Polynomial functions like this one are very versatile and can model a wide range of scenarios in mathematics, economics, and the sciences due to their smooth and predictable curves.