Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=x+2, g(x)=x^{4}+x^{2}-3 x-4$$

Step-by-Step Solution

Answer

The compositions are (a) \(x^4 + x^2 - 3x - 2\), (b) \(x^4 + 8x^3 + 25x^2 + 33x + 10\), (c) \(x + 4\), with domains \(x \in \mathbb{R}\).

Step 1: Determine (f ∘ g)(x)

The composition \((f \circ g)(x)\) implies substituting \(g(x)\) into \(f(x)\). With \(f(x) = x + 2\) and \(g(x) = x^4 + x^2 - 3x - 4\), \((f \circ g)(x) = f(g(x)) = g(x) + 2 = x^4 + x^2 - 3x - 4 + 2 = x^4 + x^2 - 3x - 2\).

Step 2: Find Domain of (f ∘ g)(x)

The domain of \((f \circ g)(x) = x^4 + x^2 - 3x - 2\) is all real numbers since it is a polynomial, which are defined for all \(x \in \mathbb{R}\).

Step 3: Determine (g ∘ f)(x)

The composition \((g \circ f)(x)\) involves substituting \(f(x)\) into \(g(x)\). Thus, \((g \circ f)(x) = g(f(x)) = (f(x))^4 + (f(x))^2 - 3(f(x)) - 4\). Replacing \(f(x)\) with \((x + 2)\), we have:\((x + 2)^4 + (x + 2)^2 - 3(x + 2) - 4\).

Step 4: Simplify (g ∘ f)(x)

Start by expanding \((x + 2)^4\) and \((x + 2)^2\). The expressions are: \((x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16\) and \((x + 2)^2 = x^2 + 4x + 4\). So, \((g \circ f)(x) = x^4 + 8x^3 + 24x^2 + 32x + 16 + x^2 + 4x + 4 - 3x - 6 - 4\) which simplifies to \(x^4 + 8x^3 + 25x^2 + 33x + 10\).

Step 5: Find Domain of (g ∘ f)(x)

The domain of \((g \circ f)(x) = x^4 + 8x^3 + 25x^2 + 33x + 10\) is all real numbers since it is a polynomial, which are defined for all \(x \in \mathbb{R}\).

Step 6: Determine (f ∘ f)(x)

The composition \((f \circ f)(x)\) requires substituting \(f(x)\) into itself. Thus, \((f \circ f)(x) = f(f(x)) = (x + 2) + 2 = x + 4\).

Step 7: Find the Domain of (f ∘ f)(x)

The domain of \((f \circ f)(x) = x + 4\) is all real numbers as it is a linear function, defined for all \(x \in \mathbb{R}\).

Key Concepts

Understanding Polynomials

Polynomials are expressions that include variables raised to whole-number exponents, along with coefficients. For example, the function \(g(x) = x^4 + x^2 - 3x - 4\) is a polynomial. Polynomials can take many forms, such as quadratic, cubic, or quartic, depending on the highest power of the variable.

Here are some key characteristics:

• Degree: The highest exponent of the polynomial. In \(g(x)\), the highest exponent is 4, so it's a quartic polynomial.
• Terms: Consist of variables and coefficients. Each term is of the form \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer exponent.
• Operation: You can add, subtract, multiply, and even compose polynomials, which means you can substitute one polynomial into another.

Polynomials are smooth and continuous functions and their graphs are smooth curves, without any breaks. This makes them predictable and easy to work with across the entire real number line.

Exploring the Domain of a Function

The domain of a function is the set of all possible inputs (x-values) for the function. It essentially answers the question 'What can you plug into this function?' Without running into issues like dividing by zero or taking the square root of a negative number.

For polynomials:

• They have a domain of all real numbers \(\mathbb{R}\). This means polynomials are defined everywhere along the real number line, from \(-\infty\) to \(\infty\).
• For example, both \((f \circ g)(x) = x^4 + x^2 - 3x - 2\) and \((g \circ f)(x) = x^4 + 8x^3 + 25x^2 + 33x + 10\) are polynomials. Hence, they are defined for all real numbers.

Understanding the domain is crucial, especially when dealing with compositions, as it helps identify valid inputs for the resulting function. For linear functions, which we'll explore next, the domain is similarly all real numbers.

Linear Functions and Their Simplicity

Linear functions are the simplest type of polynomial functions. They have the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. In our example, the function \(f(x) = x + 2\) is a linear function.

Here are some features of linear functions:

• Slope: The constant \(m\) represents the slope of the line. It tells us how steep the line is, or the rate of change of the function.
• Y-intercept: The constant \(b\) is the y-intercept, indicating where the line crosses the y-axis.
• Graph: When graphed, linear functions produce straight lines, which are predictable and easy to interpret.

Linear functions also have a domain of all real numbers, \(\mathbb{R}\), since there's no restriction on \(x\) values. They also have a relatively simple composition, as demonstrated in \((f \circ f)(x) = x + 4\). This simplicity makes them valuable tools in understanding more complex function compositions.