Problem 75 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The functions are \((f \circ g)(x) = 2^{x+1}\) and \((g \circ f)(x) = 2^x + 1\), both with the domain \((- fty, fty)\).

Step 1: Define Composition of Functions

The composition of two functions, denoted as \((f \circ g)(x)\), is defined by substituting the output \(g(x)\) into the function \(f(x)\). Similarly, \((g \circ f)(x)\) is defined by substituting the output \(f(x)\) into \(g(x)\).

Step 2: Solve for \((f \circ g)(x)\)

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

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

The function \((f \circ g)(x) = 2^{x+1}\) is defined for all real numbers since the exponential function \(2^x\) is defined for all real numbers. Thus, the domain of \(f \circ g\) is all real numbers \((-fty, fty)\).

Step 4: Solve for \((g \circ f)(x)\)

To find \((g \circ f)(x)\), substitute \(f(x) = 2^x\) into \(g(x) = x+1\).\[(g \circ f)(x) = g(f(x)) = g(2^x) = 2^x + 1.\]

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

The function \((g \circ f)(x) = 2^x + 1\) is defined for all real numbers since the exponential function \(2^x\) is defined for all real numbers. Thus, the domain of \(g \circ f(x)\) is all real numbers \((-fty, fty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial because it tells us what inputs a function can accept. Simply put, the domain includes all the possible real numbers that you can put into a function that lead to a valid output. For example, for a function like \(f(x) = \sqrt{x}\), the domain includes all non-negative real numbers since you cannot take the square root of a negative number and remain in the set of real numbers.

When we talk about the domain of compositions like \((f \circ g)(x)\) or \((g \circ f)(x)\), we essentially focus on the intersection of domains of individual functions involved in the composition. This means:

• Check the domain of each function before composing.
• Ensure the output from one function can be an input into the other function.

In our case with \(f(x) = 2^x\) and \(g(x) = x+1\), both functions can accept all real numbers as inputs. Hence, the domains for both \((f \circ g)(x)\) and \((g \circ f)(x)\) are all real numbers \((-\infty, \infty)\). Think of real numbers as any number you can think of on the number line, from negative infinity to positive infinity.

Exponential Functions

Exponential functions are a special type of function where the variable, often represented as \(x\), is in the exponent. The general form of an exponential function is \(f(x) = a^x\), where \(a\) is a positive constant, other than one. These functions are crucial in many areas, including finance, biology, and physics.

Some characteristics of exponential functions include:

• They grow very quickly as the value of \(x\) increases.
• They have a constant base (\(a\)), which determines how steep the curve is.
• The domain includes all real numbers, \((-\infty, \infty)\), meaning you can raise a positive number to any power.

In our functions, \(f(x) = 2^x\) is an exponential function. When we compose it with \(g(x) = x+1\), like in \((f \circ g)(x) = 2^{x+1}\), the behavior still remains exponential. This means the function will still slope upwards away from zero as \(x\) increases.

Real Numbers

Real numbers are the backbone of algebra and calculus. They include all the numbers you are most familiar with, such as:

• Integers: ...\(-3, -2, -1, 0, 1, 2, 3\),...
• Rational numbers: such as \(\frac{1}{2}, \frac{3}{4}, -1.5\)
• Irrational numbers: numbers that cannot be expressed as a simple fraction like \(\pi, \sqrt{2}\)

When a function like \(f(x) = 2^x\) or \(g(x) = x+1\) has a domain of real numbers, it means it can accept any number from this set, providing the function is defined for those inputs.

For the compositions \((f \circ g)(x)\) and \((g \circ f)(x)\), having a domain of all real numbers indicates they are quite flexible. You can input any real number to get a result without any restrictions, which is not always the case for other types of functions, like logarithmic or square root functions. This makes exponential functions particularly useful in modeling a variety of real-world situations without domain limitations.