Problem 33 - Algebra and Trigonometry

Question

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=2 x+3, \quad g(x)=4 x-1 $$

Step-by-Step Solution

Answer

\(f \circ g(x) = 8x + 1\), \(g \circ f(x) = 8x + 11\), \(f \circ f(x) = 4x + 9\), \(g \circ g(x) = 16x - 5\). All domains are all real numbers.

Step 1: Understand Function Composition

Function composition involves applying one function to the results of another. We denote the composition of two functions, \(f\) and \(g\), as \((f \circ g)(x) = f(g(x))\) and \((g \circ f)(x) = g(f(x))\). In these expressions, \(f\) and \(g\) are functions.

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

To find \(f \circ g\), substitute \(g(x) = 4x - 1\) into \(f(x) = 2x + 3\). So, \(f(g(x)) = f(4x - 1) = 2(4x - 1) + 3\). Simplifying, we get \(f(g(x)) = 8x - 2 + 3 = 8x + 1\). The function \(f \circ g(x) = 8x + 1\). Since both \(f(x)\) and \(g(x)\) are defined for all real \(x\), the domain of \(f \circ g\) is all real numbers.

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

To find \(g \circ f\), substitute \(f(x) = 2x + 3\) into \(g(x) = 4x - 1\). So, \(g(f(x)) = g(2x + 3) = 4(2x + 3) - 1\). Simplifying, we get \(g(f(x)) = 8x + 12 - 1 = 8x + 11\). The function \(g \circ f(x) = 8x + 11\). Since both \(f(x)\) and \(g(x)\) are defined for all real \(x\), the domain of \(g \circ f\) is all real numbers.

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

To find \(f \circ f\), substitute \(f(x) = 2x + 3\) back into itself. So, \(f(f(x)) = f(2x + 3) = 2(2x + 3) + 3\). This simplifies to \(f(f(x)) = 4x + 6 + 3 = 4x + 9\). The function \(f \circ f(x) = 4x + 9\). The domain of \(f \circ f\) is all real numbers, as \(f(x)\) is defined for all real \(x\).

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

To find \(g \circ g\), substitute \(g(x) = 4x - 1\) back into itself. So, \(g(g(x)) = g(4x - 1) = 4(4x - 1) - 1\). This simplifies to \(g(g(x)) = 16x - 4 - 1 = 16x - 5\). The function \(g \circ g(x) = 16x - 5\). The domain of \(g \circ g\) is all real numbers, as \(g(x)\) is defined for all real \(x\).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values, often denoted as "x," for which the function is defined. For a function like the ones we're dealing with here, the domain often includes all real numbers or a specific set of numbers.

• When considering linear functions, like our example functions, these are defined anywhere on the real number line.
• There are no excluded values, that is, there are no values that would make the function undefined, such as divisions by zero or square roots of negative numbers in these polynomial functions.
• Therefore, when both functions in a composition are defined for all real numbers, so is their composition.

Always ensure to check the formulas, as more complex functions might have restrictions on the domain.

Polynomial Function

A polynomial function is a function that can be expressed in the form of a polynomial. For example, the function \(f(x) = 2x + 3\) is a linear polynomial, specifically a first-degree polynomial.

• Polynomial functions can be classified by their degree, the highest power of the variable present in the polynomial. Linear functions have a degree of one.
• The coefficients, such as 2 and 3 in \(f(x) = 2x + 3\), give us the shape and position of the polynomial on the coordinate plane.
• Polynomials are continuous functions, meaning they have no breaks, jumps, or holes in their graphs.

In function compositions like \(f \circ g\), we are also effectively creating a new polynomial of potentially higher degree by substituting one polynomial into another.

Real Numbers

The set of real numbers includes all the rational and irrational numbers. It's essentially any value that can correspond to a point on an infinite number line.

• Rational numbers are fractions or decimals with a finite or repeating pattern, like 1/2 or 0.333...
• Irrational numbers can't be written as simple fractions, like \(\pi\) or the square root of 2.
• Real numbers are crucial in understanding domains because they describe every possible input we can consider for these types of polynomial functions.

For any polynomial functions and their compositions, if there are no divisions by zero or operations that lead to complex numbers, then the domain typically includes all real numbers.