Problem 72 - College Algebra with Modeling and Visualization

Question

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=a x^{2}, a>0, \quad g(x)=\sqrt{a x}, a>0 $$

Step-by-Step Solution

Answer

(f \circ g)(x) = a^2 x, x \geq 0; (g \circ f)(x) = ax, \text{ all reals}; (f \circ f)(x) = a^3 x^4, \text{ all reals}.

Step 1: Find the Composition \\(f \circ g\\)(x)

The composition \(f \circ g\)(x)\ is defined as applying \(f\) to the output of \(g\). Start by substituting \(g(x) = \sqrt{ax}\) into \(f\), where the function \(f\) is \(f(x) = ax^2\). This gives \(f(\sqrt{ax}) = a(\sqrt{ax})^2\). Simplifying this expression, we find:\[f \circ g(x) = a \cdot (ax) = a^2 x\]The domain of \(g(x)\) is \(x \geq 0\) since it involves a square root, and then \(f(x)\) can handle all real numbers. Thus, the domain of \(f \circ g\) is \(x \geq 0\).

Step 2: Find the Composition \\(g \circ f\\)(x)

The composition \(g \circ f\)(x)\ is defined as applying \(g\) to the output of \(f\). Start by substituting \(f(x) = ax^2\) into \(g\), where the function \(g\) is \(g(x) = \sqrt{ax}\). This gives \(g(ax^2) = \sqrt{a(ax^2)}\). Simplifying this expression, we find:\[g \circ f(x) = \sqrt{a^2x^2} = ax\]The domain of \(f(x)\) is \(-\infty, \infty\). Since \(g(x) = \sqrt{ax}\) only accepts non-negative \(ax^2\), \(x^2\) is always non-negative and so the domain of \(g \circ f\) is all real numbers.

Step 3: Find the Composition \\(f \circ f\\)(x)

The composition \(f \circ f\)(x)\ is defined as applying \(f\) to itself. We replace \(x\) in \(f(x) = ax^2\) with \(f(x)\), so we compute \(f(ax^2) = a(ax^2)^2\).Simplifying this expression yields:\[f \circ f(x) = a(a^2x^4) = a^3x^4\]Since \(f(x) = ax^2\) can take all real numbers, the domain of \(f \circ f\) is also all real numbers.

Key Concepts

Domain of Functions

Understanding the domain of functions is crucial for correctly determining whether a function will produce valid outputs for given inputs. In simpler terms, the domain of a function is the set of all possible input values (often referred to as "x" values) that allow the function to provide an output without encountering issues like division by zero or calculating the square root of a negative number.
For the functions given in the original problem, we have:

• For the function \(g(x) = \sqrt{ax}\), the domain is \(x \geq 0\). This is because the square root function is only defined for non-negative numbers. If you try to input a negative number into the square root, it becomes an imaginary or complex number, which we're not considering here.
• On the other hand, the function \(f(x) = ax^2\) is a polynomial and is defined for all real numbers. This means you can input any real number, positive or negative, and the function will still provide a valid output.

In compositions of functions like \((f \circ g)(x)\), it is important to respect the domains of both functions involved. For instance, when we apply \(f\) to the output of \(g\), the domain of \(g\) determines the domain of the composed function \((f \circ g)(x)\). For this composition, the domain remains \(x \geq 0\) as dictated by \(g(x)\).

Square Root Function

The square root function, typically written as \(\sqrt{x}\), is widely encountered in mathematics. Its defining characteristic is that it only produces valid outputs for non-negative inputs (i.e., numbers equal to or greater than zero). This constraint is due to the fact that no real number squared results in a negative number. Thus, trying to find the square root of a negative number leads us into the realm of imaginary or complex numbers.
Now, consider our specific function \(g(x) = \sqrt{ax}\). Here, the factor \(a\) is given as positive, so the expression under the square root is non-negative when \(x \geq 0\). This defines the input values or the domain for the function \(g(x)\) as all non-negative numbers or \([0, \infty)\).
Moreover, when calculating compositions such as \((f \circ g)(x)\) or \((g \circ f)(x)\), the nature of the square root helps ensure we're only plugging valid numbers into further functions. For example, in the composition of \(g \circ f\), the use of \(ax^2\) ensures the value inside the square root is always non-negative since squaring a real number (positive or negative) results in a non-negative value.

Polynomial Functions

Polynomial functions are among the most well-understood and familiar functions in mathematics. These functions are defined as expressions involving sums and products of variables raised to whole number exponents. A general polynomial function can be written as \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\).
In our exercise, the function \(f(x) = ax^2\) is a simple polynomial known as a quadratic polynomial. Since it's expressed with powers of the variables that are whole numbers and the coefficient \(a\) is positive, it is defined and produces outputs for all real numbers. This makes its domain \((-\infty, \infty)\).
Polynomials like this one are useful in compositions due to their predictability over all real numbers. For instance, when applied to themselves, such as in \((f \circ f)(x) = a^3x^4\), the result remains a polynomial. Thus, the domain is unchanged, and inputs can be anything within the set of real numbers.