Question

Find the domains of \((f / g)(x)\) and \((g / f)(x)\) for the functions \(f(x)=\sqrt{x}\) and \(g(x)=\sqrt{9-x^{2}}\) Why do the two domains differ?

Short answer

The domain of \((f / g)(x)\) is \(-3 < x < 3\) and the domain of \((g / f)(x)\) is \(-3 \leq x < 0\) and \(0 < x \leq 3\). The difference in the domains arises because of the constraint that we cannot divide by zero, which limits the domain based on the function in the denominator.

Step-by-step solution

Finding the domain of \(f(x)\) and \(g(x)\)

The domain of a function is the set of all possible input values (often represented by \(x\)) which will output a real number. For \(f(x)=\sqrt{x}\), the domain is \(x \geq 0\). For \(g(x)= \sqrt{9-x^{2}}\), the domain results from ensuring the expression under the square root should be nonnegative. Hence, \(9-x^{2} \geq 0\) which gives \(-3 \leq x \leq 3\).

Finding the domain of \((f / g)(x)\)

When a function is divided by another function, as in \((f / g)(x)\), the domain is the intersection of the domains of both the numerator and denominator with an additional requirement. That requirement is that the function in the denominator (in this case \(g(x)\)) should not be zero. For the given functions, \((f / g)(x)\) is undefined when \(g(x) = 0\), which occurs when \(x=-3\) or \(x=3\). So, the domain of \((f / g)(x)\) is \(0 \leq x < 3\) and \(-3 < x < 3\). We exclude -3 and 3 because at these points, \(g(x) = 0\). So, the domain is \(-3 < x < 3\).

Finding the domain of \((g / f)(x)\)

Similarly, the domain of \((g / f)(x)\) is determined by both the domains of \(g(x)\) and \(f(x)\) with the additional constraint that the denominator (now \(f(x)\)) cannot be equal to zero. \(f(x)\) equals 0 only when \(x=0\). Thus, the resulting domain is \(-3 \leq x < 0\) and \(0 < x \leq 3\).

Why the domains differ

The reason the domains of \((f / g)(x)\) and \((g / f)(x)\) differ is that different function is in the denominator for each one. In \((f / g)(x)\), we need to ensure \(g(x)\) is not zero, and in \((g / f)(x)\), we need to ensure \(f(x)\) is not zero. These constraints create differing conditions for \(x\), and hence the domains are different.

Key concepts

Domain of a Function

The domain of a function encompasses all the potential input values, commonly referred to as x, that result in real number outputs when substituted into the function. In essence, it’s a collection of permissible x values that can smoothly operate within the function without leading to undefined or non-real results.

Understanding the domain is crucial as it prevents the misuse of a function in equations or further computations. For instance, the domain of a square root function such as f(x) = \(\sqrt{x}\) is restricted to non-negative numbers, as square roots of negative numbers are not real numbers but complex numbers, which are beyond the scope of this scenario.

Square Root Functions

Square root functions, like f(x) = \(\sqrt{x}\), make us ponder which x values are reasonable. For all real numbers x, the square root of x exists only if x is zero or positive. Hence, the domain for any square root function is x ≥ 0.

When examining g(x) = \(\sqrt{9-x^2}\), it's evident that the expression inside the square root must not be negative for the function to be defined. This leads to the constraint 9 - x^2 ≥ 0, which can also be thought of as the square root function’s comfort zone, since outside of this boundary, the function starts producing results that are not suitable, in the context of real numbers.

Function Division

Diving deeper into the division of functions, as performed in (f / g)(x) or (g / f)(x), we navigate additional domain restrictions. Here, the intersection of the domains of each function must be considered, alongside the critical rule that the denominator must never be zero—since division by zero is undefined.

For example, with (f / g)(x), the domain is where both f(x) and g(x) are defined and where g(x) ≠ 0. It becomes a balancing act of satisfying both these conditions, which isn’t always straightforward and often limits the domain more than the individual functions do on their own. When we invert which function is on top, as in (g / f)(x), the domain will generally shift to reflect the new role of each function.

Real Numbers

Real numbers are the bread and butter of our numerical understanding, including all the numbers on the number line: positive numbers, negative numbers, zero, and decimals — but not imaginary numbers. The term 'real' essentially means that these numbers are not 'imaginary', which is a different kind of number that can result from taking the square root of a negative number.

The relevance of real numbers to the domains of functions is that domains are typically expressed in terms of intervals on the real number line. Hence, the domain of any square root function or a quotient of functions must lie within the realm of real numbers, which imposes some of the constraints we see when finding the domains of composite functions such as (f / g)(x) and (g / f)(x).