Problem 31 - Algebra and Trigonometry

Question

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{2}, \quad g(x)=x+1 $$

Step-by-Step Solution

Answer

\((f \circ g)(x) = x^2 + 2x + 1\); \((g \circ f)(x) = x^2 + 1\); \((f \circ f)(x) = x^4\); \((g \circ g)(x) = x + 2\). All have domains of all real numbers.

Step 1: Understand Composite Functions

Composite functions involve applying one function to the result of another function. The notation \((f \circ g)(x)\) means \(f(g(x))\). Similarly, \((g \circ f)(x)\) stands for \(g(f(x))\), and so on for other combinations.

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

Calculate \((f \circ g)(x)\):\[(f \circ g)(x) = f(g(x)) = f(x+1)\]Substitute \(g(x) = x + 1\) into \(f(x) = x^2\):\[f(x+1) = (x+1)^2 = x^2 + 2x + 1\]Thus, \((f \circ g)(x) = x^2 + 2x + 1\).

Step 3: Find \(g \circ f\)

Calculate \((g \circ f)(x)\):\[(g \circ f)(x) = g(f(x)) = g(x^2)\]Substitute \(f(x) = x^2\) into \(g(x) = x + 1\):\[g(x^2) = x^2 + 1\]Thus, \((g \circ f)(x) = x^2 + 1\).

Step 4: Find \(f \circ f\)

Calculate \((f \circ f)(x)\):\[(f \circ f)(x) = f(f(x)) = f(x^2)\]Substitute \(f(x) = x^2\) again into \(f(x) = x^2\):\[f(x^2) = (x^2)^2 = x^4\]Thus, \((f \circ f)(x) = x^4\).

Step 5: Find \(g \circ g\)

Calculate \((g \circ g)(x)\):\[(g \circ g)(x) = g(g(x)) = g(x+1)\]Substitute \(g(x) = x+1\) into \(g(x) = x+1\):\[g(x+1) = (x + 1) + 1 = x + 2\]Thus, \((g \circ g)(x) = x + 2\).

Step 6: Determine Domains

Since both \(f(x) = x^2\) and \(g(x) = x+1\) are defined for all real numbers, their composites are also defined for all real numbers. Therefore, the domains of \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\) are all real numbers.

Key Concepts

Function Composition

Function composition allows us to combine two functions in a nested manner. Think of it like putting one function inside another. Suppose we have two functions, where you apply the first one to your input and then use the output as the input for the second function. This is visually represented as

• \((f \circ g)(x) = f(g(x))\), meaning you apply \(g(x)\) first and then \(f\).
• Similarly, \((g \circ f)(x) = g(f(x))\) involves applying \(f(x)\) first, followed by \(g\).

Function composition is crucial in many mathematical processes because it allows more complex operations by combining simpler ones. Remember that the order in which you compose functions significantly affects the result.
For instance, for the functions \(f(x)=x^2\) and \(g(x)=x+1\), the compositions are different:

• \((f \circ g)(x)\) ends up as \(x^2 + 2x + 1\)
• while \((g \circ f)(x)\) is \(x^2 + 1\).

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. For any simple polynomial functions like \(f(x) = x^2\) or \(g(x) = x + 1\), the domain consists of all real numbers, denoted as

• \((-\infty, +\infty)\).

This means you can plug any real number into these functions and expect a result without mathematical errors such as division by zero or finding the square root of a negative number. Consequently, the domain for compositions like \((f \circ g)(x)\), \((g \circ f)(x)\), and others is also all real numbers because the individual functions are each defined on all real numbers.Understanding the domain is essential as it tells you the valid inputs for your function, preventing you from smacking into theoretical walls!

Quadratic Functions

Quadratic functions are a specific type of polynomial function characterized by the term \(x^2\). These functions are expressed in the standard form:

• \(f(x) = ax^2 + bx + c\).

In our context, \(f(x) = x^2\) is a simple quadratic function where its graph is a parabola opening upwards. Quadratics are widely used in various fields due to their unique properties, such as their vertex representing a maximum or minimum point on the graph.
Composing a quadratic function with another can make things interesting. When you compose \(f(x) = x^2\) with itself, as in \((f \circ f)(x)\), you get \(x^4\), a quartic function. Quadratics have solutions that are relatively easy to predict and handle, making them perfect for academic exercises designed to strengthen understanding around composition, domains, and operations.

Addition of Functions

The addition of functions is a straightforward concept where you simply add the output of two functions together. Given two functions, \(f(x)\) and \(g(x)\), their sum is represented as

• \((f+g)(x) = f(x) + g(x)\).

Function addition maintains the independent behaviors of each function while combining their effects. While our main exercise focuses on composition rather than addition, knowing that you can also sum functions simplifies understanding multiple functional operations.
Keep in mind that when you add functions, the domains of the resulting function \((f+g)(x)\) will be the intersection of \(f(x)\) and \(g(x)\) domains. However, in our problem where both functions have domains of all real numbers, their addition similarly retains all real numbers as its domain.