Question

Find \((f g)(x)\) and \(\left(\frac{f}{g}\right)(x)\) and state the domain of each. Then evaluate \(f g\) and \(\frac{f}{g}\) for the given value of \(x\). $$ f(x)=4 x, g(x)=9 x^{1 / 2} ; x=9 $$

Short answer

The product function \((f g)(x) = 36x^{3/2}\) with domain \(x \geq 0\), and the quotient function \(\left(\frac{f}{g}\right)(x) = \frac{4}{9} * x^{1 / 2}\) with domain \(x > 0\). At \(x=9\), the product function is \(1080\) and the quotient function is \(1.333\).

Step-by-step solution

Compute the Product Function

The product function, denoted as \((f g)(x)\), is found by multiplying the two functions together. In this case, the product function is: \((f g)(x) = f(x) * g(x) = 4x * (9x^{1 / 2}) = 36x^{3/2}\). Thus the product function \((f g)(x) = 36x^{3/2}\).

Determine the Domain of the Product Function

The domain of \((f g)(x)\) is the set of real numbers for which the function is defined. For this function, we must ensure that we only include real numbers such that \(x^{3/2}\) is real. So, the domain is any \(x \geq 0\).

Compute the Quotient Function

The quotient function, denoted as \(\left(\frac{f}{g}\right)(x)\), is found by dividing the first function by the second. In this case, the quotient function is: \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{4x}{9x^{1 / 2}} = \frac{4}{9} * x^{1/2}\). Thus the quotient function \(\left(\frac{f}{g}\right)(x) = \frac{4}{9} * x^{1/2}\).

Determine the Domain of the Quotient Function

The domain of \(\left(\frac{f}{g}\right)(x)\) is the set of real numbers for which the function is defined. For this function, we must ensure that \(9x^{1 / 2}\) in the denominator is never equal to zero - so the domain is any \(x \gt 0\).

Evaluate the Functions at x=9

Evaluate the functions \((f g)(x)\) and \(\left(\frac{f}{g}\right)(x)\) at \(x=9\).\n For \(f g\), \((f g)(9) = 36 * (9^{3 / 2})=1080\).\n For \(f/g\), it is \(\left(\frac{f}{g}\right)(9) = \frac{4}{9} * (9^{1 / 2}) =1.333\).

Key concepts

Product of Functions

When we talk about the product of functions, we are referring to the result of multiplying two functions together. It's similar to multiplying numbers but with a twist: functions have variables and we need to take into account their behavior across different values. For instance, given two functions, such as f(x) = 4x and g(x) = 9x^{1/2}, their product is (fg)(x) = f(x) * g(x) = 4x * 9x^{1/2} = 36x^{3/2}.
In algebra, this is comparable to combining two recipes where changes in the ingredient (in our case, x) alter the final outcome. When evaluating this combined recipe at a specific point, say x=9, we directly substitute the value into our product function and calculate the result, giving us (fg)(9) = 36 * (9^{3/2}) = 1080. It is essential, however, to rule out any ingredient that doesn't make sense—in mathematical terms, this means a value that makes the function undefined, which leads us to exploring the domain next.

Quotient of Functions

Just like how we can multiply functions, we can also divide them. The quotient of functions is found by dividing one function by another. Given f(x) = 4x and g(x) = 9x^{1/2} again, their quotient is (f/g)(x) = f(x) / g(x) = 4x / (9x^{1/2}) = (4/9)x^{1/2}.
Imagine having a piece of string representing f(x) and another measuring g(x). The quotient tells us how many times one string fits into the other. When we evaluate the quotient at x=9, we are checking the length ratio at this particular point: (f/g)(9) = (4/9) * (9^{1/2}) = 1.333. It's crucial that our denominator string g(x) never measures zero; otherwise, we'd be trying to fit our f(x) string into nothingness, which isn't possible. Hence, the domain of our quotient function avoids values that make g(x) equal to zero.

Function Domain

With any function in algebra, it's critical to determine its domain, or the set of all possible inputs. For example, the domain for the product function (fg)(x) = 36x^{3/2} includes all non-negative numbers, since we can't take the square root of a negative number without leaving the real numbers. The domain here is described as all x ≥ 0.
For the quotient function (f/g)(x), with g(x) in the denominator, the domain excludes values where g(x) equals zero, to prevent division by zero. In our case, since g(x) involves a square root, the domain is similarly x > 0, ensuring that we only deal with positive numbers. Understanding the domain is like knowing the boundaries of a playground; it tells us where we can safely play (or calculate) and where we should not step to avoid problems.

Function Evaluation

The process of function evaluation is like plugging a value into a machine and seeing what comes out. We take a number, often labeled x, insert it into our function, and perform the calculations dictated by the function. Evaluating the product and quotient functions at x=9 means using this specific value in place of the variable x.
For the product fg, this yields (fg)(9), and we perform the multiplication to see that 36 * (9^{3/2}) equals 1080. Meanwhile, for the quotient f/g, we end up with (f/g)(9) = (4/9) * (9^{1/2}) = 1.333. Through function evaluation, we can understand how our functions behave at specific points, offering us a glimpse into their larger patterns and behaviors as the value of x changes.